# Chapter 6 – Watch entire Chapter!

>>Chapter 6, More on Future Value and Present
Value. In the previous Chapter, Chapter 5, what we were doing was working with a single
cash flow, we would be interested in finding its future value or its present value and
also some other additional things like the number of years or the interest rate. In Chapter
6 we are going to be doing essentially exactly the same calculations, finding the future
value, the present value, as well as the dollar amount of the cash flow and the interest rate,
except we will be working with many cash flows. And then later in the Chapter we will introduce
special cases in which cash flows are identical rather than different values. Let’s start
with this example, you deposit \$500 in a bank today and \$600 in one year into the same bank
account. The bank pays a 9% interest annually to its depositors. Question, how much will
you have in two years in your bank account? What’s going on here is that we no longer
have a single cash flow for which we need to find the future value, we have two cash
flows. We have \$500 deposited now and \$600 deposited in one year and we need to find
their combined future value two years from today. The way we would solve something like
this is break it into individual cash flows. First, find the future value of the first
cash flow, \$500 deposited now, what will its future value be after two years? If you use
the formula for future value from Chapter 5 you would do \$500 multiplied by (1 plus
0.09) raised to the second power, that would give you \$594.05. And you can also do it in
the financial calculator if you use \$500 set it to negative, save as PV, then 9 for IY,
2 for N, and compute FV. Then separately you will do a similar thing for your second cash
flow, \$600 deposited in one year, what is its future value? First, notice that we can
no longer use the future value of the two years like we did for \$500 deposited today,
we have to use one year for the correct number of time periods, that’s because the number
of time periods what it really means is over how many years your money will be earning
interest and growing. For \$600 there’s only one year for that, and so using the formula
for future value we would do \$600 multiplied by (1 plus 0.09) which would give us \$654.
Or in the financial calculator in no particular order you enter the present value, the interest
rate, and the number of years. For example, you can do \$600, change it to negative PV,
then 9 IY, then 1 N, and then compute FV. Once you’ve done future values for your individual
cash flows then you simply combine them, you find the total future value, you combine \$594.05
and \$654 and the answer to this problem is at the end of two years you will have a total
of \$1,248.05, that’s from two deposits made at different points of time. To summarize,
in the financial calculator or using the future value formula you find individual future values,
one cash flow at a time, an then you add up the individual future values. Unfortunately,
you cannot solve something like this in one step. Example, you are offered an investment
that will pay you \$200 at the end of the first year, \$400 at the end of the second year,
\$600 at the end of the third year, and \$800 at the end of the fourth year. You can earn
a 12% return on similar investments. What is the most you should pay for this investment
today? How do you approach problems like this? Essentially, it’s saying that, it’s implying
that you should not want to invest more than what you’re getting back in the future, right?
We will be getting back four different dollar amounts in the future over four different
years. The wrong thing to do is to simply add up \$200, \$400, \$600 and \$800. In finance
kind of one of the basic rules you learn over time, which is not even written in the books,
is that you can never add things up that occur at different points of time. You need to sort
of, you know, add up apples rather than apples and oranges and so on. So how do we make them
in a way that we can correctly add them up? We need to, for example, bring them all to
year zero, discount back to today, and that’s really what the question is asking us to do.
We want to evaluate this investment opportunity today, today is when we are making the decision,
invest or not invest, right? So in today’s dollars how much is this whole investment
worth? How much are we getting back total in today’s dollars, evaluated in today’s dollars?
In other words, we need to find the present value of multiple cashes, right? This problem
is on finding the present value rather than the future value. You can imagine a timeline
under which you can put the numbers that are given. We have a lot of information that’s
given. Under year one, which is at the end of year one, you can put \$200. Under year
two \$400. Under year three \$600. Under year four \$800. And you can place the question
mark under year zero, which is today. In other words, you need to find how much this is all
worth today. Well, just like with the future value model of cash flows let’s find the present
value of model cash flows by doing the calculations we know from Chapter 5, where we were working
with single cash flows. So let’s break it all into pieces. We have four pieces here,
four different dollar amounts. Let’s first look at \$200 and ignore the rest. What is
the present value today of next year’s \$200, right? In other words, receiving \$200 in the
year is just like receiving something today, right? What is the equivalent of \$200? That
would be today’s dollar amount instead, right? Following the present value formula from Chapter
5 you would take \$200 and divide it by 1 plus the interest rate, the interest rate in this
problem is 12%, so you can divide by 1 plus 0.12 or simplify it to 1.12. The answer you
get is \$178.57. In other words, what this means is receiving \$200 back from this investment
in one year is just like receiving \$178.57 today. You can also do it in the financial
calculator by pressing \$200, making it negative, the plus-minus key, then you press the FV
key to tell the calculator that this number is my future value. Then you press 12 and
IY key, so you have just told the calculator that 12 is my interest rate per year, and
then you press 1 N, so you just told the calculator that 1 is the number of years in the future.
Compute present value, right? The same way you find the present value of \$400 that you
would be receiving in two years. Following the present value formula from the previous
Chapter you do \$400 divided by 1 plus the interest rate to the second power, which is
1 plus 0.12, that whole thing raised to the second power, or it can be written in a shorter
way, \$400 divided by 1.12 raised to the second power. This gives you \$318.88. Again, what
\$318.88 means is receiving \$400 in two years, right? So waiting for two years to get \$400.
It is just like not waiting at all and instead getting slightly less money, you’re equally
happy with these two options, right? Option one being \$400 in two years, option two being
\$318.88 today, right? So \$318.88 is the present value of \$400 that you would need to wait
two years for to get. Again, this could be done in the financial calculator. This time
you would use 2 for your N. Then \$600 received in three years, you find its individual present
value, make sure you change the power to 3, or in the financial calculator change your
N to 3. And, lastly, you find the present value of \$800 to be received in four years.
And here the correct power would be 4 and in the financial calculator 4 would be your
N, number of years. So we have four individual present values, which essentially means receiving
\$200 after one year, \$400 after two years, \$600 after three years, and \$800 after four
years, right? It is just like receiving these four individual present values that we just
calculated today, all of them today. You are equally happy with these two options, today’s
option and the delayed option, right? So now we can answer the question, how much is receiving
these four different amounts which were given to us worth? Which would then tell us how
much, you know, what is the maximum we would be willing to invest into this investment
opportunity today. You simply add up the four individual present values and get their sum
equal to \$1,432.93, this is the answer to the problem. You would not want to invest
more than this amount. If you do you would be losing money, you would invest more than
how much you get back. If you invest less, if it’s possible, then that’s a great profit
opportunity for you. Here’s another way this problem could be solved, so actually let me
go back to the previous slide and I will bring up my financial calculator right here. So
let’s turn it on, let’s make sure everything is cleared. The way I do it is I do a full
reset, I press 2nd plus-minus, which brings up reset right above it, and then enter. So
turn it off, turn it back on, we are now sure that everything is cleared from the calculator’s
memory. Here’s how you can actually do the present value of multiple cash flows in just
one step, right? Remember you cannot do that for the future value of multiple cash flows,
but you actually can do that for the present value of multiple cash flows. One step. We’re
going to use what we call in finance cash flow keys. Press the button that means cash
flow, right, we’re going to enter our cash flows that are given. Press the button in
the second row that says C F, cash flow, start with that. What does the display show? It
shows C F, subzero, equals zero. What it means is the calculator informs you that cash flow
for year zero, which is given, is zero dollars. This is something we can keep as is if it’s
true or this is something that we can change. Should we change it? No, because there’s nothing
you receive right away, we’re not given any cash flow for today, right, just like \$200,
\$400, \$600, \$800, there’s nothing today so we’ll leave it as is but we’re going to save
it. So you press the button enter, and it will then appear on the display a little extra,
like a triangle, it basically means that it has just been saved as zero dollars or here
are zero cash flow. What do you do next? You press the down arrow key in the first row
of your calculator. what has just happened? The display now says C or 1, 0.00. This is
now cash flow, next cash flow that we would like to enter, right? We are going to be entering
essentially four cash flows. What’s cash flow in year one, so what’s the next cash flow?
That’s what we now would need to change to \$200, so you put \$200, you don’t change it
to negative like we were doing in Chapter 5 for future value or present value calculations,
you just leave it as is and you save it by pressing enter again. Now kind of imagine
you’re building this table on a piece of paper, you’re writing down several rows, right, one
row for cash flow. So you press the down arrow key again, you kind of want to go to the next
line below, but you’d expect to see cash flow 02 or something like that, CO2, but it doesn’t
tell you that yet. Instead what you see is FO1 equals 1, what do you think that could
be? F stands for frequency, how many times, so how many times in the row do we have \$200?
Only once, right, because another example could have \$200 repeated consecutively, but
we don’t have it here. So this is usually the case in a lot of our problems and so we
want to leave it at 1, so we want to tell the calculator that yes, 1, frequency of 1
is correct, let’s save it. Press enter. Now the down arrow key at the present that now
you are asked to enter your cash flow number 2, which is \$400, so you press just like you
did with \$200, you now repeat these same steps all over again. You press \$400, you don’t
change any signs, leave it as is because you can think of it being the money you receive
rather than spend, right? You press enter to save it, down arrow key, it asks you for
the frequency, again we only have \$400 once consecutively or like toggle in our problem.
You enter, you save it by pressing enter. Down arrow key again, what’s cash flow number
3, that’s \$600, repeat the same steps, \$600 enter, down. It’s only once, you press enter
again, and actually you can skip the enter there for the frequencies, you can just press
the down arrow key essentially twice in a row. Down, then cash flow number four, that’s
the last one we have, \$800. You press \$800, enter, down, and like I mentioned earlier
you can do enter down or you can just press the down arrow key one more time because this
is the default, it will be saved as the default value of 1. Cash flow 5, what is it? We have
no more cash flow 5, that’s it, we only have four numbers. What do we do next? Now after
we have just entered all the cash flows that are given to us, now the only thing that’s
missing is we want to tell the calculator that we want to apply a 12% discount rate
for every single one of these given cash flows, and then we want to find their combined present
value using that discount rate. So first we tell the calculator what it is you want to
do with these cash flows and we want to find their combined present value. Finding the
key in the second row of the calculator, which says NPV, net present value, here in this
problem it is combined present value, so press that. The display shows I equals, I is like
IY, interest per year. You need specify what the interest rate is per year, that’s 12%
in this problem. You press 12, enter, right? You don’t press IY because the calculator
already tells you that that’s going to be your interest rate, so you just press 12 and
then enter, which is what I did. And then press the down arrow key, and then press compute,
\$1,432.93, just like on the slide. Okay, and I explained the same exact steps here, so
these are the steps I went through. Okay, next example, what if we changed things a
little bit in the example we just did? What if the cash flows are paid at the beginning
of each year? In other words, we move all the numbers closer by one year, so we wouldn’t
need to wait one year to get the first cash flow, \$200, we will get it right away. As
soon as we invest money we immediately get \$200 and we would get \$400 in just a year
and so on. How would you solve this problem? Well, there are several different ways. We
can do one here at a time like we did before. What’s the present value of \$200? It’s \$200,
it’s already at present. Then we would discount \$400 back by one here, \$600 back by two years,
and \$800 back by three years. Or, right, what I did here. That’s one way of doing it and
you would get \$1,604.88, right? So that \$200, present value of \$200, it’s \$200 plus \$400,
divided by 1.12 plus \$600 divided by 1.12 squared, plus \$800 divided by 1.12 cubed to
the 3rd power. Notice how the answer we got is higher than in the previous problem where
it was only a little bit over \$1,400, why does it make sense? Because so why is it that
when you get the money sooner the present value is higher? Well, think about it, you
get the money sooner, that’s better, right? We all want to get money sooner, so that’s
the intuitive explanation, but the mathematical explanation is that we don’t need one year
of discounting applied to each of our four cash flows, right? We have used no discounting
for \$200, only one instead of the previous two years of discounting for \$400, two instead
of three years for \$600, and three instead of four years for \$800, right? Another way
of solving this problem is to realize that, well, what I have just explained, you are
removing one year worth of discounting so you’re reducing the number of years over which
we discount each of the future cash flows by one, which means the same thing as taking
the answer we have on the previous slides, for the previous version of this problem and
then bringing it into the future by one year. So that’s mathematically how we can undo one
year of discounting, so you do \$1,432.93 multiplied by 1 plus the interest rate, which is 1 plus
0.12 or simply times 1.12 and this is how you get the same result, \$1,604.88. In the
financial calculator how would you do this? So, by the way, this is where if you clear
by pressing 2nd and FV, clear the sum of money, it does not clear everything. For example,
if you press on the cash flow key and you can actually now keep pressing the down arrow
key it will show you what is still saved. See, it never got cleared, and this is where
I do a full reset of my calculator. I press 2nd, plus-minus, enter. And now if you press
the cash flow button and then start pressing the down arrow key everything is back to zeroes,
right? So everything we just did on the previous slides is gone, now we can start fresh. Okay,
how do we start fresh? We press the cash flow key, right? Cash flow zero means what do we
have in here, zero, and this time it’s no longer nothing because we have just shifted
everything closer by one year, everything is one year sooner, which makes \$200 receivable
today, so that’s our cash flow zero. Now we need to change it to \$200, press \$200, enter,
down arrow key, cash flow 01. What’s the first cash flow in the future? That’s \$400, so just
like before press \$400, save it by pressing enter, press the down arrow key. What’s the
frequency of that first cash flow in the future? \$400 is only once, it’s not repeated again
in the years, so we’ll leave it at one, and you can either press enter and the down arrow
key or just the down arrow key. The calculator will keep one for you. Cash flow 02, that’s
the second one in the future, that’s year two on our timeline. We change it to \$600,
press \$600, enter, down, leave the frequency at 1 again, press the down arrow key again.
Cash flow 03, the third one in the future, which is our fourth cash flow that’s given,
\$800. Just like before press \$800, enter, down, down. What’s cash flow 4, that would
be the fifth one that is given to us. We don’t have anything like that, we are done with
entering the cash flows. Now we want to tell the calculator I want you to give me their
combined present value at 12% discounted. You press the NPV button in the second row.
It asks you for the I, which is the interest rate. Put 12, enter, now we have just saved
it, down arrow key, compute, \$1,604.88, just like on the slide. And I reviewed the cash
flow keys in the financial calculator on my next slide, so this is exactly what I just
did. Next example, your broker calls you and tells you that he has this great investment
opportunity. If you invest \$100 today you will receive \$40 in one year and \$75 in two
years. If you require a 15% return on investments of this risk should you take the investment?
If you did not know anything about finance, present value, future value, and anything
like that you would kind of look at just the dollars that are in front of you and you might
think that, oh, I spend \$100 but I get a total of \$40 plus another \$75, which is \$115, I
get \$115 back if I just invest \$100. So at the first glance it might seem like a great
investment opportunity, but what we have learned over the last lectures is that this may turn
out to be wrong and that’s all because of the interest rate. The interest rate determines
how much the future numbers are worth to you now, right? What would be an equally, you
know, good scenario, right? So instead of waiting how much would you get now and be
equally happy? In other words, the right way to approach this problem is to look at \$40
and at \$75 and calculate how much they’re worth to you today, find their combined present
value. If the number you get is above \$100 that you would need to spend now then it’s
a great investment opportunity, if the combined present value is less than \$100 it means it’s
a bad investment opportunity because you would be spending \$100 for something that’s worth
less than that, that would not give you that money back, you would actually lose money.
Okay, so how much is receiving \$40 and \$75 worth to you today? You can also rephrase
today how much would it be? One amount that’s reflecting both \$40 and \$75 received in the
future? So \$40 will be received in one year, \$75 in two years, this is what is given, you
can find the present value of \$40 by taking \$40 and dividing by 1.15, that would give
you \$34.78. In other words, if you didn’t have to wait a year how much money received
today would make you equally happy? So you would be equally happy if you did not have
to wait a year to get \$40 and receive today \$34.78, that’s the present value of next year’s
\$40. Similarly, you find the present value of \$75. To get it you would need to wait two
years. If you did not want to wait two years how much could you receive today that would
make you equally happy? That’s finding the present value of \$75. We take \$75, divide
by 1.15 squared. Again, what’s 1.15? It’s 1 plus 0.15, which is 15% annual interest
rate in decimals. And the second part indicates how many years in the future, you know, \$75
is at. So the present value of \$75 is \$56.71. If you combine them you get the sum of \$91.49,
what does that mean? It means this investment that offers you to wait a year and get \$40,
to wait another year and get \$75, is equivalent to receiving cash right away today, immediately,
which is \$91.49. You are equally happy between getting this one amount now and the alternative
scenario, which is \$40 in one year and \$75 in two years. And, of course, we can do this
whole calculation, rather than in three steps we could do it all in one step if you use
the cash flow keys. So let me bring up the financial calculator, let’s make sure we clear
everything, second, plus-minus, enter, cash flow. Not in a year zero, you leave it at
zero, you press enter, down arrow key. That \$40 is our first cash flow in the future.
You press \$40, enter, down. The frequency of is 1, you leave it at 1, you press either
enter, down or just down. Cash flow 02, the second cash flow in the future, that’s \$75,
\$75, enter, down. It’s only once you press the down arrow key. No more cash flows. Now
you press the net present value, NPV button in the second row. I, that’s the interest
rate, that’s 15% per year, it’s given to us. You press 15, enter, down. And the last thing
you do is press compute, 91.49, \$91.49. And you already probably read the conclusion,
the answer to this problem, because you would be investing \$100, which was given to us,
today for something that gives you back an equivalent of \$91.49 today. It’s not worth
it, you would be overpaying about \$8.50, right? You should, as we say in finance, reject this
investment opportunity because you will be paying \$100 for something that is only worth
\$91.49 today. We are done with — on the previous slides we did the future value of multiple
cash flows and the present value of multiple cash flows. Now let’s look at some special
cases. We’re going to look at, we’re going to first look at the future value of multiple
cash flows which are all the same amount, all cash flows are the same dollar amounts.
This is what is known as annuity, it’s defined as a series of cash flows of an equal amount
at fixed intervals for a specified number of periods. So there are three things that
go into these definition of the term annuity. First, cash flows of equal amount, they must
all be equal. For example, \$4,000 over and over and over and over again. At fixed intervals,
it means they’re all spaced out equally let’s say once a year, so there’s nothing somewhere
in the middle after six months, you’re not skipping any here. And the third thing for
a specified number of periods, this tells us that we don’t have them forever, they stop
at some point. Let’s say we have them for 20 years, once per year, and that’s it. There
are actually a lot of annuities in real life and you may never realize that this is an
annuity. For example, a lot of you may have purchased a car using a car loan, how do car
loans work? You borrow some money that pays for your car, but in return you’ll be paying
back monthly payments to your lender. Typically the monthly payments would be the same, identical
dollar amount, so maybe \$250 a month for let’s say five years. So we have identical cash
flows, they’re once a month and everything stops after five years or 60 months. Other
loans, for example you take a loan to buy a house, a home mortgage loan, maybe it’s
for 30 years. Again, home loans require you to make the loan payments back to the bank
every month, equal amounts of money typically and it will all end at some point, let’s say
after 30 years, that would be 360 months. So 360 identical monthly payments. You are
renting an apartment, probably most of you do, you probably signed a lease, maybe for
two years, maybe for one year, maybe for six months. As long as it’s not a month-to-month
lease. Under let’s say a two-year apartment lease you are expected to make the same rent
payment every month for two years, 48 times, so same amount once a month 24 times. Another
example is maybe you’re paying for your car insurance, so maybe you pay twice a year,
so you’re making semiannual payments for your car insurance maybe for a few years and then
maybe your rates go up. Some people earn their salary monthly under some contract, some earn
biweekly, so every two weeks let’s say on a five-year contract, that’s also an annuity.
Property taxes, if the value of your house doesn’t change you will be paying the same
taxes every year, so same number once a year and for how long, until you sell your house,
that’s when it will all end. I have different examples here, and you might realize that
some examples have monthly payments, others have biweekly payments, others have semiannual
payments or annual payments, these are all annuities. In fact, the word annuity is kind
of misleading because it kind of makes you think of something annual, right, occurring
once a year. No, it doesn’t have to be once a year, as long as it’s the same time intervals,
it’s all annuities. It could be daily, weekly, biweekly, monthly, quarterly, annual, biannual,
every 10 years, et cetera, et cetera, etcetera. Let’s look at this first example of an annuity,
and we are now looking at future values. And you can probably guess what’s going to be
on later slides, we’re going to be looking at annuity present values. So let’s start
with annuity future value example. You want to deposit \$4,000 at the end of each year
for the next 20 years in a bank account paying an 8% annual interest rate, how much will
you have in 20 years in your bank account? Well, based on what we know this is how we
would set it up. So you would imagine this timeline, you show years in the future, here’s
one, two, three, all the way to 20, and under end of each year you put \$4,000, right? So
you have \$4,000 20 times in the future, and you place the question mark under year 20,
so at the end of the 20th year how much do you have total in your bank account? How would
we calculate something like this on the earlier slides in this Chapter? You’d break it into
pieces, right? That’s what I taught you. You would first look at the first \$4,000, what
is its future value at the end of the 20th year counting from today? Well, you would
take \$4,000, multiplied by 1 plus the interest rate and raise it to the power that indicates
the number of years over which the interest will be earned. Notice that the power is 19,
it’s not 20, because one here is already elapsed, there are 19 years remaining between end of
year one and end of year 20 and this is the correct power to be used for the first cash
flow. And the interest rate was given at 8% per year, so I do \$4,000 times 1.08 raised
to the 19th power. Second \$4,000, again we find its individual value at the end of the
20th year counting from today. Between end of year two and end of year 20 there are 18
years, so this will be the power for this future value, \$4,000 times 1.08 to the 18th.
The next \$4,000 will be treated the same way for its individual future value, the power
will become 17, and so on. One before last, at the end of year 19 it only has one year
to earn interest and so we use no power at all or you can put power of 1, it’s the same
thing. And the last deposit that you will make, \$4,000 at the end of the 20th year,
it’s already where you want it to be, so it doesn’t earn any interest. You deposit, immediately
after that you check how much you have in your account, and from just these deposits
you have \$4,000, no interest on top. So you essentially broke it into 20 pieces, right,
20 individual future values, and then you add all of them up. It takes awhile, possible
to do, but it’ll probably take you I don’t know a few minutes for sure and just hope
you don’t make any mistakes somewhere. Turns out there’s a formula for annuities, which
looks like this. In our problem you would take \$4,000 for these repeating cash flow,
you would multiply it by 1 divided by the interest rate, 0.08. Then what you get is
multiplied by the next big term, in the big parenthesis we have 1 plus 0.08 interest rate
to the 20th power, and then you subtract 1. The answer would be \$183,047.86. Notice how
this is significantly more than simply taking \$4,000 and multiplying by 20, 20 times, which
is \$80,000, it’s over \$100,000 more than \$80,000. Why is that? Interest, interest on interest,
interest on interest on interest and so on. So the extra \$103,047.86 is pure interest,
nothing else, interest that you get from the bank. And you can also do it in the financial
calculator, so if you go back to our problem, let’s bring out the calculator, let’s clear
everything, 2nd plus-minus, enter. We’re going to learn a new button now and the button is
called PMT, third row, second from the last, what do you think it stands for? That’s right,
payment, that’s the dollar amount that’s repeated again and again, the annuity payment amount.
You put \$4,000, we have many of those, and we want to find their future value, combined
future value, so before you would save \$4,000 as your PV, but we’re not going to do it this
way, we’re not going to use the PV button at all because we have a whole bunch of PVs,
right, too many of those. Instead we’re going to call it payment, we press the PMT button.
Before you press the PMT button change \$4,000 to a negative \$4,000 by pressing plus-minus
key, and then press PMT, so now the calculator knows that you have \$4,000 or whatever the
units are and that’s going to be repeated a number of times, it’s an annuity problem.
How many times does it repeat? 20 times, you press 20 and then you press N, so here the
interpretation of N is slightly different from the way it was used in the previous Chapter
to find the future value or the present value. Here N means how many repeated payments you
have because technically \$4,000 earns interest 19 times, so that’s the right N we used for
the first \$4,000. The second \$4,000 we will be earning interest 18 times, so that would
be the right and the way we know what N is from before. So we cannot like put a whole
bunch of N’s, like 19, 18, 17 and so on. Instead N when we are working with the PMT button
N changes its meaning to how many repeating numbers we have. So we put 20 N, then the
interest rate, 8 IY as always, that’s it, we don’t need any more information. We have
\$4,000 repeating every year, the annual rate is 8% and we have a total of 20 of those repeating
cash flows. Then we press compute FV and this is the number we got earlier, right? So this
is exactly the same result, \$183,047.86, that’s in dollars. And the steps I just showed in
the financial calculator are all repeated here. The key is the new key we have just
learned, and N now means how many cash flows we have, interest rate just stays as is. This
is the general formula, it finds the future value of an annuity with T cash flows, evaluated
at the end of the T period, so you take the annuity payment which is labeled as C, stands
for cash flow, multiplied by the fraction 1 over R, then multiplied by (1 plus R raised
to the T 4, then you subtract 1). The way this formula is written is it kind of uses
this multiplication factor, the cash flow that’s repeating times something else, and
this something else is a complicated term with fractions, powers in parentheses is known
as the future value, interest factor on an annuity, or FV, I of A for some interest rate,
R, and number of cash flows, T. And again there’s a table at the end of the textbook,
Appendix 4, which gives you the value for this whole future value interest factor of
an annuity for different combinations of interest rate, R, and number of cash flows, T. But
I’m not using it in class because financial calculators are a lot easier, faster, and
more flexible, they allow for more cash flows than what that table shows and it allows for
a wider range of interest rates, like partial interest rates, like let’s say 5.1%. In the
table you would only see 5% or 6%, but nothing in between. Let’s do another example, suppose
we begin saving for your retirement by depositing \$2,000 per year in a retirement account. These
days the best rate you can get is around 3% per year. How much will we have in 40 years?
So actually let’s speak on this, it’s something that I found for the investment accounts.
So that’s actually very recent summary of interest rates, February 2019, and if you
scroll down it gives you different banks and what they offer if you deposit money for a
long term and you would see that for a longer number of years that the money gets deposited
for the interest rate is somewhere around 3%, a little bit more, a little bit less.
So let’s make it kind of easier and say 3%, which is what I did. So you’re putting in
\$2,000 per year in the retirement account, you will expect to earn 3% every year on the
money that you have, and the question is how much will you have in this retirement account
after 40 years? This question is asking you to find the future value value of this annuity
with 40 identical annuity payments in the amount of \$2,000 each. So following the formula
to find the future value of an annuity you do \$2,000 multiplied by 1 divided by 0.03,
the interest rate, times (1 plus 0.03 raised to the 40th power minus 1). This gives you
\$2,000 times the entire second complicated term is 75.4, like the multiplication factor.
You can, for example, find it in Table 4 at the end of the textbook or calculate the whole
thing by hand or use a calculator that allows for parentheses and all that. The answer is
\$150,802.52. And in the financial calculator, let me bring it up, you would do the following.
Let’s clear everything just to make sure nothing gets messed up. So \$2,000 is going to be repeating,
when you know something is repeating that’s PMT, the payment, \$2,000, make it negative,
PMT key. The interest rate is 3%, 3 IY. We have a total of 40 identical deposits, so
you put 40 N, compute, FV, future value, \$150,802.52. Again notice how much of a difference the
interest makes. So if you didn’t know this formula the easiest way you would be trying
to solve it is taking \$2,000 multiplying by 40, that would be what? That would be \$80,000,
right, \$80,000. We have almost double that, we have \$150,802.52. So the rest is all interest
earned over the years. Another example, you decided to start saving and depositing \$1,000
in the end of each month. The best checking account rate you can find is 2% a year. Actually
let’s click on this link I found. Bankrate.com, again we have different banks, like a lot
of online banks, and it shows the interest rate you earn on checking accounts, so we
have 2.02%, 1.25%, and then very low percentages, they’re actually ranked highest to lowest.
So the highest is 2.02%, right? Let’s make it easier and use 2%, so 2% per year. Your
bank will be adding interest to your account monthly, you will be making these deposits,
right, \$1,000 at the end of each month for exactly two years until you graduate from
Cappelli. How much money will you have in your bank account three years from today?
This is actually a pretty complicated problem. First, let’s see what’s going on, let’s draw
our imaginary timeline that goes into the future and there are three years in the future
that are important. For the first two years or 24 months you will be making the deposits
in the month of \$1,000 each at the end of each month. So at the end of the first month
is where you put the first \$1,000, at the end of the 24th month you put the last one,
we have a total of 24 of these \$1,000 amounts. The first step would be to find the future
value of this annuity at the end of it, so at the end of the second year, which is when
you stop making any further deposits. So how much do you have at the end of 24 months?
And then because the question asks us to calculate how much you’ll have in your account after
one extra year you would then need to take the answer you get in step one and find its
future value, you treated this answer from step one as a single cash, kind of going back
to Chapter 5, right? And you find its future value after one more year, except you should
keep in mind that the interest is earned monthly so you would need to find the future value
after additional 12 months. So your end would be not one, but 12. So in first step, in the
first step we want to find how much money we’ll have in our account at the end of the
second year, which means at the end of 24 months. So 24 is what you’re going to use
for T in the future value of an annuity formula. Because deposits will be made monthly everything
should be reflecting the monthly frequency in our formula. For example, the original
number of cash flows should be 24, the interest rate should also be monthly, so find a monthly
interest rate you take 2% annual rate and divide by 12 and you get 0.1667%. This is
a little bit rounded because it’s actually 0.1666 and it just keeps going, it’s 6’s.
When you use this monthly rate and 24 for the power in the future value of an annuity
formula you will get \$24,465.67 and, of course, you can do it in the financial calculator.
How do you do that? Let’s see, I already cleared everything, but let me do it again. \$1,000
is going to be repeating, make it negative and save it as payment, PMT key. How many
times will it repeat, 24 times, put 24 N. You need to use the monthly interest rate,
which is 0.1667, we rounded it a little bit, right? And you press IY, compute future value,
\$24,065.77. On my slide it’s 67 cents rather than 77, on my slide the number is more accurate
because I used Excel to do the math and Excel kind of doesn’t round numbers in intermediate
steps, so the Excel actually used the more accurate interest rate where it’s 6’s, never
ending 6’s. Then the second step, what do you do in the second step? In the second step
you, let’s say you get kind of lazy, you’re like, oh, I know that in the second step this
would be my present value so why don’t I just save myself some time and right away press
the plus-minus button, so make it negative, and then PV. So now I’m at the beginning of
step two of my calculations. Then I do 12 months, the money sits in the account for
12 more months, 12 N, and keeps earning the monthly interest of 0.1667%, press IY, compute
future value. Oops, I’m really off, this is not what the slide shows, I got \$37,070.33
but the slide showed that I should have \$24,959.50. Which one is right? The financial calculator
made a mistake. What’s on my slide? Ah, the lower number is actually correct, so what
did I do wrong with the calculator? The mistake is that I did not clear the calculator’s memory.
I should have cleared the calculator’s memory and then did step two. In other words, I shouldn’t
have done it the lazy way. And what happened is that the calculator thought that I continue
having \$1,000 deposits at the end of each month during the third year, instead of two,
but this is not what’s going on, right? My PMTs were essentially zero during the third
year, but the calculator used the \$1,000 for the payments. And so what I should have done
is I should have cleared everything and then started from the beginning. So at the beginning
of the third year, we are now starting step two calculations, we have \$24,465.67, negative
PV. Twelve months later, 12 N, at the monthly interest rate of 0.1667%, press IY key, how
much we would have? Compute the future value. Now the answer matches my slide’s answer,
except again there are a few cents by which our answer in the calculator is off, that’s
again the rounding error because it’s using the rounded monthly interest rate. And on
my slides the number with 50 cents is actually obtained through Excel, where nothing is rounded.
So this is more accurate. In other words, before doing step two in the financial calculator
you need to clear the time, either second FV or full reset, second plus-minus N, either
way would do the trick.>>Another example. You’re saving for a new house and you put
\$10,000 per year in an account paying 8%. How much will you have at the end of three
years? A, the payments are made at the end of each year. B, the payments are made at
the beginning of each year. Let’s see if it makes any difference, and if so, how much
of a difference. Cases A and B are illustrated here. Again, each has its own little timeline.
In Case A, where the payments are made at the end of the year, the first \$10,000 is
at the end of Year One, and then we have two more after that. In Case B, with payments
at the beginning of each year, everything is one year sooner, right? So if between zero
and one we have the first year, then the beginning of the first year is under zero on my timeline.
This is where we have the first \$10,000. Everything is sooner in Case B. In both cases, A and
B, we are asked to find the future value. How much you will have at the end of three
years. So we need to find the combined future value of these three numbers at the end of
Year Three in both cases, A and B. Okay, so let’s calculate Case A future value. Following
the standard formula that we have many times, you would use 8% for the interest rate per
year, which is given, and 3 for the power in the formula. And 3 indicates how many repeated
cash flows we have. So we are making three deposits, and that’s going to be the power
in the annuity future value formula. The answer we get is 32,464. And, of course, we can do
it in the financial calculator. First, turn the calculator on. Let’s clear everything,
which is always recommended. Second, plus, minus, enter. \$10,000, make it negative, press
payment, PMT. Three is how many of those payments we have, so I press 3 N. And the interest
rate is 8%. I press 8 IY. And I’m going to compute future value. \$32,464 even. Right?
This is the same calculations we were doing in all previous problems. And now the second
case, Case B. Let’s see what’s going on here. Everything is one year sooner, right? Do you
think it makes a difference for the combined future value? Of course. Every single one
of these \$10,000 cash flows will earn interest over an extra year. So the very first deposit
will sit in the account for a full one, two, three years, as opposed to Case A, in which
the very first \$10,000 will earn interest over only one, two years. And so on. And so
a much easier way to find the future value for Case B is to take the answer from Case
A, which we already have, and add in one year worth of interest. There is one extra year
over which the cash flows earn interest, every single cash flow in our annuity. So we take
\$32,464 and multiply it by one plus the interest rate, which is one plus 8% or simply times
1.08. And we get \$35,061. This answer is higher than the answer in Part A, because of the
extra one year worth of interest that all three of our deposits will earn. And what
we did in this step, in the financial calculator, the way you would do this step is, let’s clear,
let’s clear everything first. You would take the answer from Part A, which is behind the
calculator. That’s \$32,464, make it negative, and set it as our present value, PV. One year
later, right? What’s the future value? 1 N at 8% annual interest rate, 8 IY compute future
value. 35,061 and actually there are 12 more cents in the answer. In finance, we refer
to Cases A and B as an ordinary annuity and annuity due, respectively. An ordinary annuity
is like an ordinary case, the standard, most typical case that we deal with. For example,
a lot of problems don’t actually specify when the payments are made or received, at the
end or the beginning of which time period. So if it’s not specified, the default is at
the end, and that’s the ordinary case. That’s an ordinary annuity. But the wording might
give you a hint that the payments are at the beginning of the year. So maybe they say something
like the first payment is going to take place immediately, or the first payment in our annuity
is today. Or it could say payments are made at the beginning of each year, kind of very
clearly. That’s an annuity due. So the first payment is due immediately. And in the calculations,
you see here the way we adjust for the beginning of the year is we take the regular annuity
future value formula and multiply it by one plus the interest rate. So we are adding one
more year worth of interest to every single cash flow. So just to review, annuities can
be called ordinary or due. Ordinary annuity and annuity due. Ordinary annuity has payments
at the end of each period, and annuity due has its payments at the beginning of each
time period. You can also rephrase and say that in an ordinary annuity, the first cash
flow occurs with a delay. So you have to wait one time period before you have to pay or
receive the first payment, depending on the problem. And in an annuity due, the very first
cash flow in that annuity is immediate. And this is the general formula which shows how
we can go from an ordinary annuity future value to an annuity due future value. You
simply multiply by one plus the interest rate. And so you can say that the future value of
an annuity due is always going to be a bigger number than the future value for an ordinary
annuity. That’s because of this one plus R term. But there is another way to find the
future value of an annuity due. In the problem we just did, we can do the following. Let’s
clear everything. We can tell the calculator that the calculations I’m about to do for
an annuity are going to be based on an annuity due. And you press four keys. Second payment,
it brings up the text right above the key payment that says BGN, begin, and then second
enter. So basically you’re saying begin set, set to the begin mode. And then you can even
turn the calculator off and back on. Now the display shows three extra letters, BGN, begin.
Everything is now in the begin mode. And now, after setting the calculator to the begin
mode, I want you to do what you did for an ordinary annuity. You did what? You did 10,000,
make it negative, BMT. We have three of those. 3M. 8% is the annual interest rate, 8 IY compute.
Future value, what do we get? We do not get the answer for Case A. We get the answer for
Case B, annuity due answer. \$35,061.12. So this is probably going to be your favorite
method, right? So to summarize, there are how many methods do we have? I would say three.
One, one cash flow at a time. So the first \$10,000, find its individual future value
using three years. The second \$10,000, find its individual future value after two years.
And the last \$10,000, find its individual future value after one year. And then add
up the three individual future values. That’s one. Kind of the most time consuming method.
The second one is what I have on the slide, where you first calculate an annuity future
value, but the default in the calculator is that everything is at the end of the year
rather than at the beginning. So it will not give you an accurate number, and you need
to then, as the second step, need to make an adjustment yourself. You take the number
you got and multiply it by one plus the interest rate, 8%. So this on my slide is basically
the second method. And the third method is where you set your calculator to the begin
mode, second payment, second enter, and then do what you did for ordinary annuity. Because
the financial calculator will know that this is not an ordinary annuity that you are doing
the calculations for. It’s an annuity due. Then next, — So we already covered the set
into the due mode or begin mode, set to due or set to the begin mode in the calculator.
Now let’s look at annuities, present value. Present value of an annuity. Let’s say you
have this, a very, really situation. You’re ready to buy a new car. You are driving around,
shopping around, and you are choosing between three cars that you really like. A Nissan,
which costs \$20,000, a Hummer costs \$27,000 and a nice, bright blue Porsche for \$30,000.
So these are your options, and you would like to buy one of these cars. You also know that
you have no money at all, you’re going to be taking a loan to fully cover the cost of
whichever car you will be able to buy, to afford. But that will be determined by how
much you can afford to pay per month on the monthly car loan payments. So a typical loan
requires that you make payments back to the lender every month. And let’s say your budget
allows you to spend up to \$600 per month on car loan payments. Let’s say you have not
the best credit score. Let’s say it’s 595. This table here, I found it on ValuePenguin.com
website. It shows the relationship between what kind of FICO credit score you have, like
in which range you are and for each range, it shows what the annual interest rate would
be on new cars if the loan is for 60 months, which is five years. So what is the interest
rate for a five-year car loan, depending on your credit score? In our problem, your credit
score is 595. Where is it? It’s in the range between 590 and 619, for which the corresponding
annual car loan interest rate is 14.06%. So let’s put all this information together and
answer the question. Which one of these three cars can you afford? The question is kind
of vague. It doesn’t really tell you what you need to calculate, right? So how are we
going to approach it? Well, with your credit score of 595, the interest rate is 14.06%
per year. The loan would be for 60 months. What would be the monthly interest rate on
the loan? It will be 14.06% per year divided by 12 months. It gives 1.1717% per month.
And we did this because since the loan payments are monthly, everything in our calculations
must be monthly. Whatever calculations we’ll be doing, everything should reflect the monthly
frequency, monthly interest rate, number of months, monthly cash flow. And what we can
do is how about we calculate what the maximum loan is that you can afford. How would we
do that? We can do the following. Every month — so, okay. How should we think about loans?
What is a loan? Somebody gives you money today, somebody lends you money, a bank, some lender.
And then you will be paying it back in small pieces over many, many months, maybe even
years, many years. But the lender will charge you interest. Okay, so in a way, the loan
amount should equal the sum of all future loan payments that you will be making back
to the lender in return. Except the sum should reflect the whole [inaudible] money, so the
interest rates should be somehow embedded in our summation. And the way it’s done is
we don’t just add — we don’t just do \$600 times 60 months. We discount each of the future
monthly loan payments. We discount it back to today, using the monthly rate and the appropriate
number of months. For example, we don’t just take the first \$600 that we will be paying
in one month, and then adding the next one — the first \$600 would need to be divided
by one plus the monthly interest rate. So one plus 0.011717. The next \$600 from the
end of the second month is divided by the same thing, but raised to the second power,
and so on. And the last \$600 in this sequence of a total of 60 loan payments is divided
by the same term, one plus the interest rate, 0.011717, decimals raised to the power 60.
Okay, so this is what you need to do. Very easy, right? Well, not really. It would take
you forever. It’s very frustrating to have to calculate something like this. There are
60 terms in this sum. But don’t worry. Of course, there is a quick and easy solution
to this. I’ll show you two different shortcuts. The first one is the formula. It looks a little
bit similar to the formula we had for an annuity future value. This is an annuity present value.
The present value of all loan payments in our particular problem, which says take the
loan payment, \$600, multiply it by the reciprocal of the interest rate, so 0.011717. Then multiply
that by open parentheses, one minus reciprocal of one plus the interest rate, raised to the
power 48, closed parentheses. That gives \$25,751.77. If you go back — so just remember this number,
\$25,700 approximately. Which car can you guy for that kind of money? For that kind of money,
you can only buy the Nissan, because that’s the only price that falls under this loan
limit, which was 25,000 and a little bit. You cannot afford a Hummer, a nice blue Hummer.
You cannot afford a nice yellow Porsche, right? Unfortunately. Okay, so this is the formula
and it only covers the purchase of a Nissan. But, of course, there’s another shortcut.
You can do the whole thing in the financial calculator. And based on what we were doing
on the previous slides for annuity future value, you probably have already figured out
what I’m about to do. Let’s clear everything. Remember, the calculator was on the begin
mode. Now we need to go back to the ordinary annuity mode. And resetting the whole calculator
just takes care of that. So how do you find the present value of an annuity? The present
value. Annuity means we need to use the PMT button. What’s our payment? That’s the loan
payment that we will be making every month for the next five years or 60 months. So we
put 600, we make it negative. We save it as our payment, the PMT key. How many of those
we will have? We will have them over five years, but everything we are entering should
reflect the monthly frequency. So five years is 60 months, 60 N. And the interest rate
is also monthly, which is 1.1717, and then you press the interest per year button, I
over Y. Compute. This time, you’re pressing the present value key. \$25,751 and a little
bit. That’s only enough to buy the cheapest one of the three car choices that you have.
Right? So this is how you do it in the financial calculator. This is what I just did. There’s
actually another way of doing it. See how many different ways we can [inaudible]? There
is a way to do it in the financial calculator using the cash flow keys. Maybe I should open
up this slide and do it here. So how else can we do this? We can use the cash flow keys.
Let’s clear everything, start all over. Cash flow. What’s cash flow zero? What kind of
payments are we going to be making on the loan? We’re not going to make anything right
away. We can wait a month before we pay the first \$600. So we save it as zero dollars.
We keep it as is, we press enter, down arrow key. What’s our first cash flow? Cash flow
one, the first one in the future. That’s the first \$600 payment. You put 600, keep the
sign as positive, enter, done. What’s the frequency rate? Now you can actually change
the frequency. How about we make it 60? This is exactly how many times in a row we have
the same exact cash flow. We’re going to be paying \$600 60 times in a row, consecutively.
Same number every month over and over and over for 60 months. So when the display asks
you for the frequency, which is set to one as the default, you are going to change it
to 60, press enter. And then the down arrow key. What’s cash flow owed to? Is it \$600
in the second month? No, you actually don’t need anything anymore. You’re done. You have
just entered 60 identical cash flows. We are done with all cash flows. There is no more.
And now you press the NPV button. It asks you for the interest rate. You put the monthly
rate, which is 1.1717, enter, down, compute. Same answer. \$25,751 and a little bit, a few
cents. And again, I show both financial calculator methods on this slide. So either one is okay
to use. You can use the one with the payment button or you can use the cash flow keys.
It’s completely up to you. Everything gives the same answer. In fact, let’s check something.
So let’s go to this website. It’s Bank of America. Somewhere in the Bank of America
website you’ll find — actually let’s go to the original page. Bank of America website.
Auto loans. Click here, auto loan calculator. This is where you want to be. Total loan amount.
What was it? Total loan amount was — 25,751 is what we found, right? The term is for 60
months, that’s five years. The interest rate, this is actually per year. What was it? 14.06%
right? Calculate payment. Exactly \$600. \$600 is the required payment on a loan in the amount
of \$25,751. So what I’m trying to say is that all of these formulas we are learning in this
class, future value, present value, future value for annuity, present value for annuity.
Everything we are learning is real. There is nothing made up or simplified. This is
exactly the formulas that banks use to compute your loan payments or anything related to
loans and other things in finance. So I think it’s kind of cool that now you know what’s
behind all those loan calculators that you see online. Now you can do all that yourself.
Now you can walk to your bank, ask for a loan, bring your financial calculator, and they
will be talking to you differently. So in general, when you have an annuity, and you
need to find its present value, essentially one cash flow at a time, you discount it back
to times zero using the appropriate number of time periods. And then you’re combining
all individual present values. And the shortcut to that is the formula I will show you, which
says the present value for annuity equals the annuity cash flow multiplied by one divided
by the interest rate. Then multiplied by open parenthesis, one minus, then we are subtracting
a fraction with one on top and on the bottom one plus the interest rate and that raised
to the power of four, closed parenthesis. So the present value of an annuity gives the
value with, in today’s dollars, one time period before the first cash flow. See, there is
like a delay here. The first cash flow is not today, but it’s with a delay, with a one
time period delay. And we’re going to come back to that later, what if it happens immediately.
So this is the formula for the present value of a T period annuity. And again, there’s
a table at the end of the textbook, which gives the value for this, present value interest
factor of an annuity, which is everything in the formula but the repeated cash flow
amount itself. Four different combinations of interest rate R and time periods T. So
there is a table in the Appendix A.3 at the end of the book. Again, this is optional.
I’m not going to be using it in class. Let’s look at — again, this is optional. I’m not
going to be using this table in class. Example, let’s say a family won \$10 million in a lottery.
The money is paid in equal annual end-of-year installments of \$333,333.33 over 30 years.
By the way, I got this annual payment by simply taking \$10 million and dividing by 30 years.
The family also has an option to instead accept one lump sum payment today, which is, by the
way, how it is in real life. So next time you read about some lucky family or coworkers
alternative option of collecting money, one large payment today and no more in the future.
So in this example, how much would that one lump sum payment to be received today be?
How much should it be? And we also know that the proper discount rate in this problem is
5%. So to solve this problem, we need to find the present value of all future lottery payments,
which was Option Number One. Because an equivalent amount of money that you could collect today
should represent the sum of all future — Amounts of money, and except we cannot just
add them all up. We cannot just do \$333,333.33 times 30 of those. We need to add up their
present values. So we need to add 30 present values. And the shortcut is how we can do
it sort of in a faster way rather than finding 30 individual present values. So this is the
present value of an annuity formula. So you take the annual amount, multiply it by one
divided by 5%. The discount rate, which was given to us, then you multiply that by open
parenthesis, one minus fraction, which has one on top and on the bottom to the 30th power
we have one plus 5% interest, and then you close the big parenthesis. The answer we get
is a little over \$5 million. \$5,124,150.29 cents. So have you noticed how, in this problem,
the family won \$10 million, but if you choose Option Number Two, one lump sum collected
today, it’s going to be just a little over half of \$10 million. Everything is done correctly.
This is how things work in real life, too. So \$10 million spread over 30 years in these
equal amounts, \$333,000 and a little bit, is just like receiving just a little over
\$5 million today. You’re indifferent between the two options, collecting a little over
5 million today and, instead, collecting \$333,333.33 30 times, spread over the next 30 years. You
are indifferent between these two options. And, of course, in the financial calculator,
you can verify this answer. Let’s clear the calculator first. We need to find the present
value of an annuity, so we will be computing PV. For that we need three things, the payment
amount, the repeating cash flow amount, how many of those we have, N, and the interest
rate between all of them, which is IY. So let’s start with the payment amount. \$333,333.33
negative PMT, 5 is the interest rate, 5 IY, we have 30 of those
payments, 30 N, compute, PV. And that’s our answer, \$5,124,150.29. Just like on the slide.
Let’s say there are two investment opportunities, A and B. A pays you \$1000 at the end of each
year for 20 years. B pays you \$1000 at the beginning of each year for 20 years. The discount
rate is 10%. Which one of these two investment opportunities is more valuable in today’s
dollars, and by how much? So we need to compare today’s values of these two investment opportunities.
In other words, think of it as comparing an equivalent of receiving \$1000 20 times, but
an equivalent of that would be receiving one large amount of money today. So how much would
that one large amount of money today be equal to for Case A and Case B, and then compare
those. So, in other words, compare the two present values. So we have an annuity going
on in this problem, right? \$1000 is repeated again and again and again at the end of each
year. So it’s an annual repeating payment, and we have it 20 times. Same in Case B. How
do we find the present value of the first annuity in Case A, where the payments are
at the end of each year? We already know the formula from before, where we used \$1000 for
the payment, 10% or 0.1 in decimals for the interest rate throughout the formula, and
24 T, the number of payments. That gives us \$8513.56. Of course, we can use the financial
calculator. Let’s move it here. First clear everything. 1000 is our payment, so 1000 negative
PMT. We have 20 of those, 20 N. And the interest rate every year is 10%, 10 IY. Compute, present
value. \$8513.56. Okay, now what about Case B? In the second case, if you compare it to
Case A, the only difference is that since everything is at the beginning of each year,
everything starts sooner. Everything is one year sooner. The first \$1000 in this investment
opportunity will be received by you today. You wouldn’t need to wait for it for one year.
The second \$1000 will be received in one year, whereas in Case A, you would need to wait
for two years for that. Right? Everything is sooner in Case B. Do you value this investment
opportunity in Case B higher or lower? I would definitely value it higher, because it’s sooner.
I get all my money sooner. I don’t need to wait for it. Waiting kind of costs money,
essentially. And the way we should look at how that will affect our numerical answer
is we don’t need to discount those \$1000 cash flows so much. We want to undo one year worth
of discount for every single one of them. They’re not going to be discounted by one,
two, three, all the way to 20 years. They’re going to be discounted by zero, one, two,
all the way to 19 years. There’s one year less of discounting for each of these 20 \$1000
cash flows. And mathematically to undo the discounting, we can multiply the answer we
have in Case A by one plus the interest rate. So it’s like we are taking \$8513.56 and bringing
the 8 into the future by one here. Because really, if you compare the time lines, Case
B is like — it’s as if we just moved the question mark in Case A into the future by
one here. This is where we want to be, where the first cash flow occurs. And so it’s like
we want to be in the future by one here, and that’s this adjustment we are doing to our
calculations. So the fastest way to get the answer would be taking the answer from Case
A and multiplying by one plus the 10% interest. And we get a larger number, \$9364.92. Intuitively,
the larger number in Case B reflects the fact that we like this investment opportunity better.
We get our money sooner by one here. Who wouldn’t like that? And in the financial calculator,
this step we just did could be done by — let’s clear it all first — by finding the future
value of a single cash flow. So I guess in the formula I put here, it kind of looks like
we are finding the present value, which is correct. But we are finding it as the future
value of the present value we have in Case A after one year. So it might look a little
bit confusing here. Okay, so the present value of the 20 identical future cash flows in today’s
dollars can be found by taking the answer from Part A, which is \$8513.56, make it negative,
that’s your PV. And we want to bring it into the future by one here. So this way, we are
sort of adding back one year of discount. And we’re undoing it. We are undoing one year
worth of discounting. So I press 1 N. The interest rate is 10%. I press 10 IY. And I’m
computing future value. \$9,364.92. And then you can compare the two results in Cases A
and B, and the difference between the two present values that we found is \$851. So the
opportunity to receive the money sooner by one year is more valuable by \$851. Now just
like with annuity future values, annuity present values also have two formulas, one for an
ordinary annuity and one for an annuity due. Same interpretation as before. An ordinary
annuity is sort of the default case when the cash flows are all at the end of each time
period. For example, at the end of each year. And annuity due is when the cash flows are
due immediately. So they are all at the beginning of each time period. And the transition from
an ordinary annuity present value to an annuity due present value is done by this multiplication
by one plus the interest rate. And so the present value for an annuity due is always
bigger than the present value of an ordinary annuity. And something important. This was
exactly the same for the future value of an annuity due versus the future value for an
ordinary annuity. With future values, it’s the same result. An annuity due future value
is also always bigger than an ordinary annuity future value. So it goes for — it’s true
for both, present value and future value. And, again, there is a way to tell the calculator
that the calculations I’m about to do are going to be for an annuity due case. So you
can set your calculator to the annuity due mode or the so-called begin mode. If you go
back to this slide, to Cases A and B, then here’s what we can do. So let’s reset the
calculator first. Everything is cleared. Then I want to tell the calculator that I’m setting
you to the begin mode or annuity due mode. Press four buttons, second, payment, second,
enter. If you turn it off and back on, you will see the small letters BGN, begin. The
calculator’s now in the begin mode. And then, try to follow the steps that we did in Part
A for an ordinary annuity and see what you get. We did \$1000 for our payment amount,
we made it negative, we pressed the PMT button. Then, because we have 20 of those, we pressed
20 N, the interest rate was 10% per year, so we pressed 10 IY, and with this information,
we wanted to compute the present value. But look at what the display gives you. It doesn’t
give you the answer for Case A. It gives you the answer to Case B, 9364.92. And that’s
because you told the calculator before you put in all these numbers that were given that
we are going to be now in the begin mode. So you can switch between the end mod and
begin mode very easily by pressing second, payment, second, enter, which is also explained
on the bottom of Slide 42. So you set the calculator to the begin mode, and then you
do the calculations like for an ordinary annuity. But that gives you the answer for an annuity
due. If you want to go back, then you can either press second, payment, second, enter
again, and that will take you back to the ordinary annuity case. Or I just simply reset
my calculator. It works even faster. Let me show you the cash flow keys using the most
recent problem with Cases A and B. So let’s look at Case B, an annuity due. Clear everything.
Actually let’s start with Case A. What’s going on in Case A? How can you do it using cash
flow keys? Cash flow, cash flow zero is nothing, so you leave it as is, press enter, down arrow
key. Then we have \$1000, right? 1000, enter, down. What’s the frequency of it? It’s not
one, which is the default, it’s 20. You put 20, enter, down. There is no more cash flow
that needs to be entered. You are done with all 20 of them. Then you press NPV. It asks
you for the I, the interest rate. You put 10. We have a 10% rate. 10, enter, down, compute.
\$8513.56. Now what will be different for Case B, an annuity due, with the \$1000 payments
occurring at the beginning of each year? Make sure you clear the calculator. It’s important.
Cash flow key. What’s cash flow zero? It’s no longer zero dollars. Now the first \$1000
occurs immediately, and that’s what you need to put in the calculator, 1000, enter, down.
What’s cash flow one? The first one in the future. It’s \$1000 in one year. You put 1000
again, enter, down. What’s the frequency of it? How many times in a row will we have it?
The right number is 19, because the 20th, we already took care of that with cash flow
zero. There are 19 more left in the future, one through 19. So this is the right frequency.
Enter, down, no more. You press NPV. The discount rate is 10%, 10, enter, down, compute. \$9364.92.
So maybe this is a little bit tricky in finance that often times there is no single way to
solve a problem. Or maybe it’s a good thing, because some of you prefer one method, others
find a different method more intuitive. So this way, there will be the right method for
each one of you. Let’s summarize annuity present value and annuity future value, like some
effects of things that go into the calculations. Both annuity future value and annuity present
value will be higher if we increase the dollar amount of the cash flow that’s appearing.
It probably makes intuitive sense, because really, future value and present value is
nothing but a sum. So if the numbers you’re adding up are larger, of course, the sum will
be larger, too. And that’s true for the case when we want to find the future value or the
present value. The second thing that they have in common is the number of cash flows.
Again, if you’re increasing the number of terms in the sum, the answer will be higher.
So both the future value and the present value will be increasing if you increase the number
of cash flows. So that’s N in the financial calculator. One thing that makes them different
is the interest rate, or the discount rate. The R in the formula, or the IY in the financial
calculator. The future value is higher if we increase the interest rate. The present
value decreases if we increase the interest rate. And the last item I have to compare
the two annuities is annuity due versus ordinary annuity. And this is, again, something that
both future value and present value of an annuity have in common. Both annuity future
value and annuity present value are higher when they’re based on an annuity due case,
rather than on an ordinary annuity case. Let’s do a mid-chapter summary. So in the first
part of Chapter Six, we talked about first how we would do the math for many different
cash flows, so multiple cash flows, as opposed to a single cash flow, which was the case
in the previous chapter, Chapter Five. To find the future value, you can split it all
into small pieces, one cash flow at a time, find its individual future value, and then
add all of them up. To find the present value, you can do the same thing. You can find individual
present values and then add them up. Or you can use cash flow keys in the financial calculator.
Then we looked at many different cash flows which have identical values. That’s what an
annuity is. We looked at the future value of an ordinary annuity, where you can calculate
individual future values and add them up, except that might take you a lot of time if
it’s a very long annuity. A much faster solution would be using the PMT key. You would enter
PMT and N, IY and compute the future value. To find the future value for annuity due,
you can do it in the financial calculator in the same one step, using the PMT key, except
you first set the calculator to the annuity due mode. As for the present value, for an
ordinary annuity, the longest way would be finding individual present values, one for
each cash flow, and then adding them up. A much faster way is using the PMT button, where
you enter the payment, the number of payments, and the interest rate, IY, and computing PV.
Or you can use cash flow keys in the financial calculator. And to find the present value
of an annuity due, you can do the same exact things, except if you choose the method with
the PMT button, you first need to set your calculator to the begin mode. And if you want
to use the cash flow keys, then the very first cash flow for year zero should not be zero,
but whatever the cash flow amount is.>>Now we will look at our next topic in Chapter
6. What if we know the annuity value, either the present value or the future value? But
instead what we don’t know is, for example, the cash flow amount, the repeating dollar
amount. That’s C in your formula or PMT button in the financial calculator. Or maybe instead
we do know the cash flow amount, but we don’t know the interest rate which is R in the formula
or IY in the financial calculator. Or maybe instead what are unknown is the number of.
The repeat in annuity cash flows, which is T in the formula and N in the financial calculator.
So this part of the chapter is a little bit tricky, just to warn you. If we are working
with an annuity present value, let’s say we know the loan amount which is always the present
value of all future loan payments back to the lender. Right? So we know the loan amount,
right? But now, and this is the formula that we have for the annuity present value, C is
the cash flow amount which repeating, R is the interest rate, and T is how many repeating
cash flows we have. Or maybe it’s a present value of an annuity due problem. So maybe
it’s a loan where payments are due on the first of each month, something like that.
Then the formula has an extra multiplied by 1 plus R, at the end of it. In either case,
if you need to find for the cash flow amount, then we can mathematically rearrange it and
take present [inaudible] of an annuity or annuity due and divide by everything that’s
the annuity payment is multiplied by. So we get this division. Or we can use the financial
calculator where you would be computing PMT. Then maybe instead, we know the cash flow
amount. We know the annuity present value or annuity due present value, but you don’t
know the interest rate. There is no formula that allows to easily solve for the interest
rate. So you do a trial and error. The easiest way would be using the financial calculator
where you would be computing IY. And, of course, there’s a third thing that could be instead
unknown. That’s the number of annuity payments. That’s T in the formula or N in the financial
calculator. Again there is no easy way to rearrange the formula. So you either do trial
and error or you can use the financial calculator where you would be solving for N. Then this
slide has basically the same things, but it’s like a different setup. It’s set up in which
we know the annuity future value or the annuity due future value. In this case, there is a
formula to compute the annuity payment where you divide the future value of your annuity
or annuity due by everything that the annuity payment is multiplied by in the original future
value of an annuity formula. Or in the financial calculator to solve for an annuity payment,
you would be computing PMT. Then to find the interest rate of the number of annuity payments,
the easiest would be using the financial calculator and solving for IY, which is the interest
rate, or N which is the number of annuity payments. So let’s think of, you know, examples.
So let’s now do a few examples. And they will be basically, you know, all over the place.
In some we know the annuity present value, in others the annuity future value. Some are
annuities due. Some are ordinary annuities. In some we need to find the annuity payment,
in others the interest rate, and then yet in others the number of annuity payments.
So everything will be kind of completely messed, mixed up. Well, messed up too. So let’s start
with this first example. You won a lottery, 12 million dollars. You have two choices.
First you take \$7,042,000 today and nothing else in the future. Just one single lump-sum
payment to you. The second choice you have is you take 26 annual payments of \$461,538
each. Where did this strange number come from? That’s a very simple division of 12 million
dollars by 26. The question is, what is the implied interest rate? I would say for problems
like this, you know, kind of don’t rush into trying to use a calculator and solving for
something. I would say first answer these four questions. First, probably the most important
one, are we working with the present value or the future value of an annuity? Which one
is known? The second one, is it an ordinary annuity or an annuity due? Then what are we
asked to solve for? And once you figure out what you need to solve for, then you need
to figure out what it is that you would need
to input in your calculator. So there are always three numbers that they’re given, and
the fourth one that you need to solve for. So is it an annuity present value or an annuity
future value problem? The money you want, 12 million dollars, one way to collect it
is to receive 1/26th of that, of 12 million dollars every year for the next 26 years.
Right? So 12 million dollars that you won is basically the sum of the annual installments
in today’s dollars. So this is the present value of an annuity. So you kind of really
need to understand from the origin whether this large number, which is your annuity value,
is the present value or the future value. I guess here the hint is you want the money
today. So anything that’s today is the present value. Okay, so we answered the first question.
We are working with the present value of an annuity. The second question is, is it an
ordinary annuity or an annuity due? How can we tell? An ordinary annuity is when the annuity
payments, the numbers which are repeating, are occurring at the end of each time period.
For example, at the end of each year. And an annuity due would be when the payments
occur at the beginning of each year. So they’re due immediately. What do we have here? When
would the annual installments be paid? Hmm, actually it doesn’t really say. What should
we assume? The default is at the end of each year. That’s always the default in, you know,
finance classes. And that’s the ordinary case, and that’s where the name ordinary annuity
comes from. So we are working with the present value of an ordinary annuity. All payments
will occur at the end of each year. So to collect your first installment, you would
need to wait one year from today. That’s what it means. Then the third question I put here
is what do we need to solve for. Well, the question was about the implied interest rate.
So it’s pretty straightforward. We need to solve for interest rate. So that’s IY in the
financial calculator, if that’s what you will use to calculate the answer. In the formula,
that’s R which stands for rate of interest. And once you figure out what you need to solve
for, then the last question I would recommend you answer to yourself is, okay, which numbers
do I need to input in the calculator. So we already know that we’re given the present
value of an annuity. That’s one; \$7,042,000. Then the other two will be N. That’s 26 future
annual installments, if you choose the second option. And the third number that is given
is the cash flow amount, the annuity payment or PMT in the financial calculator. That’s
the amount of each annual installment for the next 26 years, \$461,538. Four hundred,
sixty-one thousand, five hundred, thirty-eight dollars. Okay, now the easiest would be probably
using the financial calculator. You input your present value N and PMT. And you’re computing,
I write. So let’s do that in the financial calculator. So here’s the financial calculator.
Turn it on. I also [inaudible] to start by clearing everything, because even if you used
your calculator last week, last time, it will store some values unless you already cleared
it. So do a full reset just to be, you know, absolutely sure nothing is stored. Second,
plus minus, answer. The only problem is, if you increase decimal places then now that
drops down to 2, which is the default number. So maybe we should increase the decimal places.
How would you do that? That’s actually always a good idea, especially in multi-step problems.
Here it’s actually okay, but and my answer is rounded as you can see to two decimal places.
But let’s say you need more than two decimal places in your answer. Then that’s the first
thing you would do in the calculator. How do we increase decimal places. Second, decimal
place on the bottom. The default decimals is 2. You want to change it, let’s say, to
6. Press 6, answer. You’re done. Now you can turn the calculator off and back on. It’s
going to be calculating everything from now on, until you reset it, to 6 decimal places.
So now let’s solve this problem. Seven million, forty-two thousand. Make it negative. That’s
our annuity present value. And you’re indifferent between collecting this one lump sum amount
today or, instead, collecting 26 annual payments in the amount 461,538. So let’s input that
like second option values. So that’s equivalent to collecting 26. So you put 26 in. The payment
amount for each installment would be 461,538. You leave the sign as is. You don’t change
it to negative. So kind of the way to remember it is everything that’s today is 1 sign. Everything
that’s in the future is with the opposite sign. So we entered our present value with
the minus sign, which means we should leave the PMT with the positive sign. So I input
the value, and then I press PMT. Now it’s stored as the annuity payment. Okay, so we
put in [inaudible] and N, PMT. And how we just press compute, IY; 4.4312 or 1. If the
answer needs to be rounded to two decimal places, then that’s 4.43 percent per year,
the annual interest rate. Now let’s look at the second problem. You want to save up \$60,000
for your graduate degree that you plan to start in four years. You keep all your money
in an account that offers an 8 percent rate, per year, with the interest earned monthly.
If you plan to set aside fixed amounts of money at the beginning of each month, what
will your monthly deposit be? So once again the same four questions. Are we working with
the present value or the future [inaudible] annuity? Is it an ordinary annuity or an annuity
due? What do we need to solve for, and which numbers are given? Technically you can add
a couple more questions at the beginning of that questions list. For example, I would
say the very first question should be are we working with a single cash flow or multiple
cashes? The way you know that this problem is not about a single cash flow, even though
it looks like there’s only one, but multiple cash flows is again from the origin. You’ll
be setting aside amounts of money at the beginning of each month. So it’s something repeated
monthly. There are many cash flows. So it’s not like a Chapter 5 type problem with one
cash flow. It’s Chapter 6 type problem with multiple cash flows. Then another extra question
you could ask yourself is now that we know that it’s multiple cash flows, are they different
amounts of the same amounts. Well, we will be setting aside fixed amounts of money at
the beginning of each month. So every month it will be the same amount of money that we
will be setting aside. So it’s the same cash flow repeating again and again. What do you
call that? That’s an annuity. Okay, now let’s go, you know, look at this list. Are we working
with the present value or with the future value of an annuity? Which one is known? So
if we want to save us \$60,000 for our graduate degree in four years, then \$60,000 is in the
future. Right? That’s our goal for the future. That’s what we want to have at the end of
all of our monthly deposits. So this is, you know, how you can understand that it’s the
future value of this loan. So \$60,000 is the future value of many, many, many equal amount
deposits. And that’s what the question is asking us to compute. So we are working with
the future value of an annuity. We’ve just answered the first question. The second question,
is it an ordinary annuity or an annuity due. So this refers to the timing of the repeated
cash flows. When do they take place? Do they take place at the end of each time period
or at the beginning of each time period? We know that we will be setting aside fixed amounts
of money at the beginning of each month. The word “beginning” is the key word for annuity
due. So we are working with the future value of an annuity due. And then the last two questions
are what is it that we need to solve for. For example, in the financial calculator,
what are we going to be computing? And then the question after that is asking what is
given. And again with annuities, three numbers are given. The fourth one is something that
we need to compute. So what do we need to solve for? We need to solve for the amount
of monthly deposit. That’s the annuity payment, PMT in the financial calculator or in the
formula that’s capital C. Which three numbers are given? We already know that we are given
the future value. And two more are how many deposits and the interest rate per time period.
So there’s a trick in this problem. Everything will be monthly, right? We are going to be
making monthly deposits. As soon as you realize what the frequency is of your annuity payments,
everything that you enter in the calculator should reflect that frequency. So, for example,
for your N, the number of deposits, you need to put how many monthly deposits you have.
Because now problem will be making deposits over four years, you should put not 4 but
4 times 12 equals 48 for your number of deposits. So our N will be 48. And for the interest
rate. We cannot use the 8 percent annual rate for IY. We need to divide it by 12 to get
the monthly rate, which is 0.66667 percent per month. That’s not in decimals. That’s
in percent. Okay, because it’s an annuity due, the first thing we need to do in the
financial calculator is tell it that we’re going to be doing the calculations for an
annuity due. So we say set the calculator to the annuity due mode or the begin mode.
Okay, let’s do that together, financial calculator. Let’s clear everything. Let’s increase the
decimal places. Let’s do four decimal places, second decimal place for answer. [Inaudible]
let’s start. How do we set the calculator to the annuity due mode or so called begin
mode? You press these four buttons. You press second. Then you press PMT. Then you press
second again, and you press answer. And now if you turn the calculator off and back on,
first it still remembers that you set it to four decimal places. And the second thing
you may notice is the small letters bgn on the display, which stands for begin. We’re
now in the begin mode. And now let’s enter the three numbers which are given and compute
PMT. So we are given the future value of an annuity due. So you just put 60,000. Change
the sign to negative and press the FV button. We have a total of 48 deposits that we would
need to make in order to reach this \$60,000 goal in our bank account. So you press 48,
N. And the last thing is the monthly interest rate. Remember everything should be monthly
in our calculations. The monthly rate is 0.66667. So typically it’s a bunch of 6s. You can just
keep pressing 66 multiple times, but I round it like this. It’s a little bit random, I
guess. After entering the monthly rate, you press the IY button. And finally you press
compute. What was it that we are computing? Payment. Compute payment, \$1,057 and approximately
72 cents. Right? If you round to cents. Next example. You have just bought a new flat-screen
TV that cost \$4,000. You fully financed the purchase, which means you took a loan for
the full amount, \$4,000. You didn’t pay anything out of your pocket. The loan agreement requires
you to make annual payments of \$750.54 at 5 percent interest with the first one due
immediately. How many loan payments will you have to make until the TV loan is paid off?
Again I’m asking you to ask yourself the same four questions as in my previous examples.
The first question is do we know the present value or the future value of an annuity. Four
thousand dollars, the loan amount, is the present value of an annuity of loan payments.
You can almost remember it as a rule. Any loan is the present value of loan payments,
right? So \$4,000 is the PV. Then the second question I have, for my next slide, is are
we working with an ordinary annuity or annuity due. We know that the first future loan payment
will be due immediately. So all loan payments will be made at the beginning of each month,
right? Once a month. Sorry, once a year. They are one year apart from each other. And if
the first one is immediate, then that right away should tell you that we’re working with
an annuity due. Then what is it that we need to solve for? How many loan payments? Clearly
that’s N in the financial calculator or T in the loan formula. And which numbers are
we going to enter in the calculator? The three that are given. Four thousand will be our
PV, 750.54 is PMT, and 5 is going to be our IY, the annual interest rate. So we are working
with the present value of an annuity due. We need to solve for the number of payments,
which is N in the financial calculator. And the numbers that we are going to enter are
PV, is \$4,000, IY is 5, and PMT is 750.54. From my experience, a lot of students somehow,
you know, tend to solve these kind of problems where it’s asking you for the number of payments
wrong. I see that a lot. So the wrong way would be kind of doing it the simplest way
that sort of comes to your mind. You take the loan amount 4,000 and divide it by annual
loan payment, 750.54. That gives 5.33 years. While this is intuitively the right way to
look at this problem, in finance this is a wrong calculation. The reason it’s wrong is
because you’re completely ignoring the so called time value of money. The whole, you
know, interest rate that the lender charges you, it’s not reflected here at all. And so
the correct way of doing it is, first, because it’s an annuity due, you press these four
keys to set your calculator to the annuity due mode. Second payment, second enter. Let’s
do that. Financial calculator, let’s clear everything. Second plus, minus, enter. To
set to the annuity due mode, you press these four buttons; second, PMT, second, enter.
Now the display shows bgn, begin mode. And enter the three numbers which are given. Four
thousand dollars is your loan amount, which is always the present value of all loan payments
that we’ll follow in the future. So you want to change your sign to a negative, and then
save it as PV. Then the interest rate is 5 percent. Five, IY. That’s per year. And the
annual interest. Sorry. The annual loan payment will be \$750.54. So you put 750.54. You leave
this sign as positive, because you already change the present value to negative 1. But
the way, either/or. Either present value’s negative and payment positive, or the other
way around. So press PMT, compute, N. Six exactly. Six payments. The first one due immediately
and five more in the future. Six payments total. Six loan payments total, and you will
fully pay off your loan balance. There will be nothing left, you know, for you to pay
to the lender. So if you did it the wrong way, you will be off by, you know, quite a
lot. If you did it the right way, you will get six years. Next example. This one is difficult.
I labeled it as difficult. It requires two steps. Let’s say high school friends decided
that two years from today they will take a graduation trip to Europe for the entire summer,
which is three months. For this trip, they decided to start saving money by depositing
all together, so two friends combined, \$200 at the end of each month into their joint
savings account that pays a 12 percent interest per year. So it means 1 percent monthly. The
rate is way too high, but it’s just in this example to make the math easier. They will
spend the entire saved amount during their summer trip in Europe. Here’s the question.
How much money, and it needs to be a fix amount, will they be able to withdraw from their joint
savings account at the beginning of each of the three months of their European travel?
So here’s what’s going on. They will be making deposits for the next two years once a month.
So a total of 24 deposits. We know that they will be depositing \$200 at the end of each
month. So the first deposit that they will make will be one month from today And the
last, the 24th deposit, will be at the end of two years, which is March 24. That’s the
first step. What are you going to calculate here? You will be able to calculate how much
money they will have by the beginning of their trip. What the problem is asking us is how
much will they then be able to withdraw at the beginning of each month during the three
months of their travel to Europe. So the first month is between end of month 24 and end of
month 25, counting from today. The beginning of that is the beginning of month 25. Right?
So it’s under 24 on my time line and so on. So you place three question marks. And each
withdrawal amount should be the same. So all three should be the same number. In the first
step you can calculate how much they will have saved by the beginning of their trip.
So you would find the future value of an ordinary annuity with 24 monthly deposits at the end
of each month. In the second step, you will then use the number from step 1 as the present
value of three monthly money withdrawals, and you’ll be solving for the annuity payment,
PMT in the financial calculator. So step 1, after two years, after making a total of 24
deposits, \$200 at the end of each month, they will be able to save \$5,394.60 if you plug
in the numbers into the future value of an ordinary annuity formula. Then, so, of course,
you can do it in the financial calculator using 200 negative as PMT, 24 for your N,
1 for your monthly rate. That’s your IY. And you’re computing future value. The wrong way
to then calculate the deposit, you know, one of three equal amount deposits, is to take
the total savings at the beginning of the trip and divide by three. Because that doesn’t
take into account the fact that once the first amount is withdrawn, everything that remains
will earn 1 percent interest over the next month. And then after the second month is
withdrawn, what’s left earns interest again over the third month. So how do we include
all of that into our second step? The correct way is to solve for annuity payment either
using the formula or the financial calculator. The important thing in the financial calculator
is you need to set it to begin mode. You need to clear it first and set it to the begin
mode. The reason you’re to clear it, is because there will no longer be any additional deposits,
you know, made during the three months after the last, you know, deposit is done. The reason
you need to set it to the begin mode is because the friends, while they’re in Europe, will
be withdrawing money at the beginning of each month. So in a way there are four steps that
you need to follow to solve this problem. Okay, so let’s do it in the financial calculator.
Let’s clear everything, second, plus, minus, enter. Step 1. How much will they have in
their savings account by the beginning of their trip? We need to compute the future
value of an ordinary annuity. The annuity payment is how much they will be depositing
each month. That’s \$200. So I press 200. I change the sign to negative, and then I press
PMT. How many deposits will they make over two years? Because the deposits are monthly,
we need to stick to the monthly frequency throughout all of our calculations and the
number of deposits should be 24, 24 N. Then we need to enter the interest rate that the
money will be earning in the bank account, you know, until the friends leave for their
trip. Again the interest rate must be monthly, which is 1 percent per month. You put 1, IY,
and you’re computing future value, \$5,394.69. Right? Four, 6, 9. Actually, you know, I have
only 6. So you can, you know, feel free to add 9 here. And then we want to clear everything.
Second, plus, minus, enter. We want to set the calculator to the begin mode, because
the next step will be an annuity due. You press four buttons; second, payment, second,
enter. And now do the second step and the last step. But the beginning of the trip,
they will have \$5,394.60, .6. Make it negative. Save it as your present value, because that’s
kind of the sum of the three money withdrawals before they are done, right? Before the money
is withdrawn. So that’s the present value in the second step. That was the future value
in the first step, but that’s the present value in the second step. Then they’re going
to be withdrawing money three times. Put 3, N. The money that remains in the bank after
each amount is withdrawn will earn a 1 percent interest every month. And we should take that
into account. So you put 1, IY, and you’re computing PMT, \$1,600. Sorry, \$1,816.12. And
my answer’s actually rounded to the whole dollar, 1,816. A pretty complicated problem.
I promise I won’t ask you anything like this on any exam, but it’s good to see how complex
things can get. And I also thought it’s a pretty realistic problem, right? Saving money
for something. And then figuring out how much you can use out of it, you know, regularly,
if that needs to be in fixed amounts. So we are done with annuities; solving for annuity
future value, annuity present value, annuity payment, annuity interest rate, and number
of annuity payments. Now we are going to look at the next topic in this chapter, perpetuities.
Something perpetual, something that never ends, keeps going. Right? Perpetuity is defined
as a perpetual series of cash flows of an equal amount at fixed intervals. Basically
it’s exactly the same thing as an annuity that continues forever. There is no end to
annuity payments, keeps going forever. Okay? Let’s say we have a perpetuity with some repeating
cash flow that keeps going forever, and we want to find the present value of the whole
endless sequence of these cash flows. If you were to do it one by one, right? Individual
present values and add them up. We would do the first cash flow in the future, discounted
by 1 here. So we divide it by 1 plus the interest rate, plus the present value of the second
cash flow two years from today. Or two time periods, more generally, from today. You take
cash flow divide by squared, 1 plus R. Right? One plus R to the power 2. Then you do the
same for the third cash flow in the future using the power 3 in the denominator, plus
the present value of the fourth cash flow in the future. Right? Where you use power
4 in the denominator. And so on, forever. Where do you stop? You never stop. You should
keep adding all those terms, right? So is there a way to simplify it? Yes, the simplest
formula you’ll ever see in finance. The present value of a perpetuity is equal to the cash
flow which is repeating forever divided by the interest rate. One thing you may notice
is that this perpetuity present value formula gives you, you know, the sum of all future
cash flows into today’s dollars one time period before the very first one. Right? So if it
started immediately, then in this formula you would need to just plus cash flow. So
you would have C divided by R, and then you’d add cash flow. But, you know, technically
you can even call perpetuity due. That’s something, right, sort of made up at some point? But
the book doesn’t even talk about anything like this. So this is pretty much, you know,
what the book says for perpetuities. The first cash flow is not immediate, but with a one
time period delay. And the present value of that and the sequence is C divided by R. What
would be an example of a perpetuity in real life? There are actually not that many real
life examples of perpetuity. There’s something called preferred stock. So some companies
have two different types of stock shares, one known as common stock where the company
would be paying dividends that are neither guaranteed nor of equal amount. So maybe today
it pays a \$1 dividend in its share. Next time it pays nothing. Then nothing again. And then
pays a \$3 dividend per share and so on. So it kind of fluctuates and is not sort of regular.
Preferred stock is another type of stock that exists for some companies. That’s when a company
promises to preferred stockholders a fixed dividend every period, usually every quarter
forever. Right? So as long as the company’s around, it will be paying the dividend to
the investors forever. And not just the dividend. It’s going to be fixed dividend. So it’s the
same dollar amount forever, which makes it a perpetuity. Let’s look at what we are given
for this problem. A company sells preferred stock for \$80 a share. So we know the price
per share. It’s offered dividend is \$3 paid once a year. The discount rate is 4 percent.
The question is, is this stock price correctly or is it overpriced or is it underpriced?
Overpriced means 480 is way too much; it should cost less than that. Underpriced means \$80
is too cheap; it should cost more than \$80. So how about we use the information that we
are given to calculate how much one share should cost? And because we are dealing with
preferred stock, which is a perpetuity, we can simply find the present value of a perpetual
stream of \$3 using 4 percent for the discount rate. So \$3 divided by 0.04, that’s 4 percent
interest rate in decimals, gives us \$75 per share. So you should not be willing to pay
more than \$75 for getting back \$3, you know, at the end of each year forever when the company
generates a 4 percent return on its money every year. What do we know? We know that
the shares are sold for \$80, right? Which was given. That’s way too much. That’s \$5
more than anybody should be willing to pay. Are the shares priced correctly? No, they’re
way too expensive. They are overpriced. They’re sold for \$80 while their true value is only
\$75. So they’re \$5 overpriced. By the way, technically, you know, we can use the financial
calculator for perpetuity problems, although you’ll probably realize that it’s not worth
it because it’s a very simple formula. You can do it by hand in just a few seconds. But,
okay, let’s say you’re curious. Can it be done in the financial calculator? Yes. Because
a perpetuity is like an annuity with endless cash flows, for your N put a few 9s, like
999, 999. I think if you put way too many 9s, it will get confused and give you an error
message. So put three 9s. It’s good enough. Or because we are solving for perpetuity present
value, which is like the present value of an annuity with many, many, many, many payments,
you can also use cash flow keys like we learned for, you know, present values of multiple
cash flows. Again here with it asks you for the frequency of your \$3 cash flow, put a
few 9s. Three 9s should be good enough. Put 999, and then when you find the net present
value it will give you pretty much \$75. So you might be off by a couple of cents. Actually
let’s try that. Let’s try using the annuity keys. Let’s clear everything. Second, plus,
minus, enter. Let’s actually increase the decimal places just to get an idea how far
off you’ll be, because of the rounding error. And why is there going to be a rounding error?
Because technically N is not 999. It’s infinity. But 999 is a good approximation. So let’s
see how far off you’ll be. Let’s increase the decimal places. Second decimal place,
let’s make it 7, enter. Our payment is \$3 every year, forever. Three. Make it negative,
press PMT. Four is the interest rate per year. Four, IY. For RN, you’re going to put three
9s, 999, 999, N, compute present value. See, we basically get \$75 and zero, you know, after
the decimal place up to 1,2, 3, 4, 5, 6, 7 decimal places. So probably in, you know,
a much higher order decimal place, we’re going to see something other than zero. But, you
know, you can see how 999 for infinity approximation is perfect. Now let’s instead try the cash
flow keys. Let’s clear everything. Let’s again increase the decimal places. Second decimal
place, let’s make it 7, enter. Cash flow keys. You start by pressing the cash flow button,
CF. You don’t get \$3 paid to you right away. So you save cash flow 4 times zero as zero
dollars, enter, down. Cash flow number 1 in the future. That’s your first \$3 dividend
that will be paid to you. You press 3, enter, down. Now it asks for the frequency. How many
times in a row are we going to have the \$3 amount? The default is 1. We need to change
it to technically infinity, but I’m going to tell you to put in just large enough number
will be fine. Let’s put 999, enter, down, MPV. The interest rate per year is 4 percent.
Four, enter, down, compute. Again we get basically \$75 even. Technically you would see numbers
other than zero. Some were in higher order decimal places, but in the first seven decimal
places it’s a zero. So it’s perfect. Nine hundred ninety-nine could be used, you know,
in place of infinity if you want to, you know, use the financial calculator whenever can
including perpetuities. My suggest would be, you know, just stick to the formula, do it
by hand. I think it’s a lot faster. Next example. You just paid \$750,000 for financial security
that will pay you and your heirs \$42,000 a year forever. What rate of return are you
earning on this policy? T The word “forever” should right away tell you that we’re working
with a perpetuity. You just paid 750,000. It happened now. So that’s the present value.
This is the present value. We already know what it is. Forty-two thousand dollars is
something that will be paid a year. Right? So repeating every year, every year forever.
So that’s the perpetuity payment. Right? So we have the present value of the perpetuity
equals the perpetuity cash flow divided by the interest rate R. We are given the perpetuity
present cash and we are given the cash flow amount. So what we need to solve for is the
rate of return. We need to rearrange this little formula to solve for R. We can solve
for the interest rate by dividing the perpetuity payment by the present value of perpetuity.
So we have just rearranged the present value of a perpetuity formula, which is C divided
by R, to solve for R. R equals C divided by present value of perpetuity. And then we just
plug in the numbers that we are given. There are only two numbers that we need to solve
for the perpetuity interest rate. Forty-two thousand divided by 750,000, which gives 5.6
percent. So that’s the annual interest rate. In general, there are only three things that
could be required to, you know, by sold for in a perpetuity problem. Either we find the
present value, which is C divided by R, or we know the present value and we know the
perpetuity payment like in the last example, and we’re asked to solve for the interest
rate R. Or we know the interest rate R. We know the perpetuity present value. And instead
we are asked to solve for the perpetuity payment. C equals PV multiplied by R. So this three
formulas are actually the same one formula, just rearranged differently. It’s really the
same one formula. The next topic is called growing annuity and growing perpetuity. This
is like a little bit more complicated. Well, probably a lot more complicated, right? Than
what we just did. So think about the cash flows in our annuity or perpetuity being not
constant, but instead growing but, you know, following some pattern. They’re growing at
the same rate every time period. For example, a growing annuity. What would a growing annuity
look like? Let’s say there are four annuity payments. So normally we would say that they
are the same amount, right? But now we are saying that the first annuity payment is this
much money. The next one is some percentage larger. So to find it, we would take C and
multiply by 1 plus G, where G is the percentage increase. Then the third will be larger again.
So we can change the power to 2, right? C multiplied by, open parenthesis, 1 plus G,
close parenthesis, second power. And then the fourth one in our growing annuity will
be calculated the same way except the power will be changed to 3. There is a way to simplify
this whole thing, and actually looks a lot like the present value of the regular annuity
except we have the growth rate in the annuity payments reflected in the formula in a couple
of places. So you would do the annuity cash flow, C, multiplied by the fraction with 1
on top and on the bottom R minus G. Then you multiply by, open parenthesis, 1 minus. The
term rates to the power T has on top 1 plus G and on the bottom 1 plus R. And you close
the parenthesis. Unfortunately you probably except me to say don’t worry, we can solve
it in the financial calculator. No, not this time. This time I’m going to tell you that
this could only be done by hand. This is probably the most complicated formula you will see
in this class. No, actually there will be one more. But, it’s okay, I’m not going to
scare you with that yet. At this point, this is the most complicated formula in this class.
And you have to use it by hand, unfortunately. Future value for growing annuity. Technically
there is a formula to find the future value for growing annuity, but it’s not in the book
and we’re going to skip it. So don’t worry about it. Present value for growing perpetuity
where the perpetuity payments are growing at the same rate over and over and over, forever.
which had C divided by R. And now the way we adjusted for the growth in the cash flow
amounts is on the bottom where you used to have R. You now have R minus G. So that’s
it. So you have C divided by R minus G on the bottom. The only little rule with this
formula is that R must be bigger than G. Otherwise let’s say R and G are equal. Then on the bottom
you have zero. And you cannot divide by zero. Or if the growth rate in cash flows is higher
than the discount rate, then on the bottom of the formula you’d have a negative number.
And dividing by a negative number would give you a negative perpetuity value, which kind
of doesn’t make sense. It’s like it cost a negative amount of money, right? So you don’t
pay something, some money for it, but instead somebody pays you to buy it. So it kind of
becomes nonsense. So R must be bigger than G. That’s the only little rule. For the purpose
of this class, it will never come up. In all problems on growing perpetuity, your R will
be larger than your G. So this issue will never come up. By the way, you might be wondering
why I never talked about the future value of a perpetuity or a growing perpetuity. That’s
because in the future. And, you know, by future it usually mean when everything end. So when
the perpetuity ends. It never ends. So if you imagine some point in time after a million
years where this perpetuity supposedly ends, even though even that is kind of a wrong thing
to say, then by that time, you know, each cash flow will earn so much interest in interest
on the interest, that each will turn into perpetuity like infinity much money. And you
have infinitely many of those infinite amounts of money. So we never actually talk about
future value of a perpetuity or a growing perpetuity. It’s just going to be an infinity.
Let’s look at this example. Your grandfather left you an inheritance that will provide
an annual income for the next ten years. You will receive the first payment one year from
now in the amount of \$3,000. Every year after that, the payment amount will increase by
6 percent. So this is how you know it’s a growing annuity. The question in this problem
is what is your inheritance worth to you today. So the question’s asking to find the present
value. If you can earn 9.5 percent on your investments. So we need to find the present
value of a growing annuity with the first annuity payment being \$3,000. And every payment
after that, so another nine, increasing by 6 percent. You need to use the formula where
you, you know, plug in the numbers and solve it by hand. So 3,000 times 1 over .095 minus
.06. Then you multiply the whole thing by, open parenthesis, 1 minus, fraction to the
10th power where, in the numerator, you have 1 plus .06 and on the bottom you have 1 plus
.095. The answer you get or should get, if you did everything right, is \$23,774. So this
is the value of this inheritance in today’s dollars. Let’s summarize, you know, what we
have learned in this chapter so far. We looked at multiple cash flows, present value and
future value. Then we looked at the special case of multiple cash flows when all of them
have the same dollar value. So they’re the same amount of money. That’s an annuity. And
annuity is defined as identical cash flows, regular frequency for limited amount of time.
We looked at how to find the future value of an annuity for an ordinary case and an
annuity due case. Then we looked at how to find the present value of an annuity, ordinary
annuity versus annuity due. Then we had several examples on computing annuity cash flow, which
is PMT in the financial calculator; annuity interest rate, which is IY in the financial
calculator. Or, instead, the number of annuity payments, which is N in the financial calculator.
A special case of annuities is a perpetuity. That’s when we have an annuity that never
ends. So cash flows keep going forever, right? They last forever. We only looked at how to
find the present value of a perpetuity, because the future value of any perpetuity would be
an infinity, an infinity large number. So we never even have examples on that case.
But we did have an example on finding the perpetuity payment when we do know the annuity
present value. Another problem would be solving for perpetuity interest rate, which is IY
in the financial calculator or R in the formula. For perpetuities, using the formula by hand,
which is PV equals C divided by R, is much easier than doing the math in the financial
calculator. In the financial calculator, what do you use for the number of payments? Technically
it’s infinitely many payments, but if you just put a few 9s, let’s say, 999, then you
will get your answer. It’s going to work. And lastly we looked at special cases of annuities
and perpetuities, which are called growing annuity, growing perpetuity. That’s when the
cash flow amount is growing at the certain rate every time period. We only looked at
the present value formulas for growing annuity and the growing perpetuity. Effective annual
rate or EAR. The idea is the following when interest payments are received more than once
a year, interest rate effectively earn over one year will be higher than the stated interest
rate. Why is that true? That’s because of the interest on the interest, the extra money
that you would be able to earn over one year. And so what we need to do is or what we’re
actually going to learn now is how we convert the interest rate per year which is given
to use, this so called stated interest rate or the quoted rate into the effective annual
rate EAR. That you know calculate — sorry. That correctly reflects all the additional
interest on the interest that’s going on. And then later we will look at how the effective
annual interest rate can be used as the correct interest rate in all the calculations which
we have just done. To find the present value or the future value, of a single cash flow
or of many cash flows or, or of annuities or perpetuities. Let’s look at the following
example. Let’s see what the effective annual rate is all about you know based on this very
simple example. You deposit \$100, the quoted annual rate is 10%. So you deposit \$100 in
your bank account now. You wait one full year until the bank transfers you your earned interest
10%. So at the end of the year you have \$110 which is calculated by taking the \$100 that
you started with and adding 10% interest to it. So 100 times open parenthesis 1 plus .1
closed parenthesis equals 110, right? In this case interest is paid annually. Once a year
you see more money in your bank account. Let’s change it a little bit. Interest is paid semiannually.
What does it mean? It means that you don’t need to wait for one full year to see more
After another 6 months you earn your partial interest again. So essentially you’re splitting
the 10% interest over one year into two parts, 5% after the first 6 months and the other
5% after the second half of year. Let’s do the math. Are we going to get \$110 after one
year? Let’s see. You deposit \$100 today how much will you have in 6 months? 5% more, right?
Which is half of the annual interest rate. And so you take 100 multiplied by 1 plus .05
and that’s how you got 105, right? \$105. Then \$105 remains in your bank account for another
half of year. So it gets reinvested for another 6 months. And now at the end of the year you
have another 5% on top of 105, not on top of \$100. So what’s the additional 5% of 105?
It’s additional \$5.25. At the end of the year we have 105 from 6 months earlier with 5%
added to it. So you have 105 times open parenthesis 1 plus .05 equals \$110.25. So with the annual
compounding of interest when you have to wait one full year to see the interest added to
you account, you get \$110 at the end of the year. With semiannual compounding you get
\$110.25. There’s additional 25 cents. It’s just little bit but it’s something. And so
the effective annual interest rate is additional you know earning 1.25% on your money rather
than you know earning just 10% which is the quoted rate. 10% and a 1/4 is the effective
annual interest rate EAR. And I also show how you could calculate \$110.25 slightly differently
where you basically say that on your initial amount of money \$100 deposited now you earn
a 5% interest once and then you earn 5% interest for the second time on top of what you had
you know half a year earlier. So \$100 times 1 plus .05 raised to the second power. In
general in order to calculate the effective annual interest rate EAR when interest is
compounded m times a year we follow this formula. E r equals open parenthesis 1 plus the fraction
with quoted interest rate on top m on the bottom closed parenthesis. The whole thing
is raised to the power m and then we subtract 1. So the fraction inside the parenthesis
which is quoted interest rate per year divided by m times per year is nothing but interest
rate per compounded interval. Another example with that 10% annual quoted rate and semiannual
compounding we earn half of the annual rate after the first half of year which is 5%.
Then m is how many times interest is earned or compounded within a year. Another example
with semiannual compounding it’s twice a year. Now let me show you how you can do this thing
in the financial calculator. Let’s turn it on. Let’s clear everything just to make sure
it’s not going to reuse something that we have stored. Second plus minus enter. Let’s
increase the decimal places. Second for decimal place. Let’s make it not 2 but 6. 6 enter.
And now let’s see how we can go from the 10% quoted annual rate to the 10.25% effective
annual rate. You do the following; second 2. It brings up the text in the second row.
That’s what they you know the key is second really means. What does we — what does it
say in the second row like right above the key with the number 2 in it? It says I-C-O-N-V
which stands for interest conversion. So we want to convert our interest from this stated
interest rate or the quoted interest rate to the effective rate. Take a look at what
the display shows. N-O-M. Press either the up or the down arrow key. Just keep pressing.
You will see that the display switched between three things; N-O-M, E-F-F, and C over y.
So go back to where it says N-O-M. N-O-M stands for nominal. It’s the same thing as quoted
or stated rate. The one that’s given, the one that, that does not reflect any additional
interest on interest going on. So you want to set the nominal interest rate in your financial
calculator to 10. Press 10 enter. It’s saved. Now press either the up or the down arrow
key right. And stop when you see c divided by y. C forward slash y. It stands for compounding
per year. What should this be changed to? With semiannual compounding or twice a year
the interest is earned, you want to set this to 2. Press 2 enter. So enter 10% quoted rate,
2 times per year is how often the interest gets compounded within a year. And now press
either the up or the down arrow key and stop when you see E-F-F on the display. It stands
for effective. All we have to do now is press the button compute. The display now shows
10.25. The effective annual interest rate is 10.25% which matches what we have on this
slide, right? If you, if your quoted interest rate is 10% then we can think of other compounded
frequencies. So we had annual once a year and semiannual twice a year. And the e r with
the 10% if it’s just once a year then the effective rate is the quoted rate. And with
semiannual compounded it’s 10.25% which we have just calculated. How about other frequencies?
Well you can think of for example earning interest quarterly once a quarter which is
4 times a year. In this case the effective rate will be 10.381%. Or monthly which is
12 times a year. The effective rate is even higher 10.4713%. And this is actually very
common in real life. A lot of things in life is monthly compounding; loans, credit cards,
rents. Most things are monthly com — compounded monthly. Then you can think of weekly compounded
or 52 times with an even higher effective rate. Daily 365 times even higher effective
rate. This special case is called continuous compounding. That’s then the frequency basically
infinitive. And in the calculator you can basically change your frequency of 2 let’s
say a bunch of 9’s. It will work or the formal formula looks like this. Exponent and then
you use the quoted rate .1 in our example. And then from the whole thing you subtract
1. This is sort of as high as it can get. 10.5171% per year. Let’s try one of these.
So let’s see once again how everything can be done in the financial calculator. Let’s
clear second plus minus enter. Let’s increase the decimal places. Second decimal place let’s
make it 6. 6 enter. And now let’s convert the quoted interest rate of 10% into the effective
annual interest rate using — which one should we do? Let’s do the monthly compounded, the
one we said is really common in real life. Monthly compounded means 12 times a year.
So after the first month you have a little bit more money. Then that larger amount stays
for another month and earns interest. And there’s now interest on interest going on.
And so on. In the financial calculator you convert using the following steps. Second
2. What’s the nominal rate? That’s the one that’s given 10%. You press 10 enter. Then
either up or down arrow key. Keep pressing. Stop when you see compounded per year c over
y. You want to change it to 12 times a year. 12 enter. Then again either up or down arrow
key. Stop when you see E-F-F effective. All you need to do now is press compute. 10.471307.
In my table I rounded to 4 decimal places for the monthly compounded so it only shows
10.4713, but it’s basically the same thing. [ Background Noise ]>>In real life I already mentioned there
are a lot of examples of where interest rate is compounded more than once a year. Car loans,
car — the payments of the car loans, the payments on home loans, payments on any kind
of loans are monthly. When you put money in the bank account, you deposit money, the bank
actually calculates daily how much you earn in interest except it waits a month before
it transfers the whole earned interest to you. So the banks actually you know all banks
actually use daily compounding of your — the money in your account. Example quoted annual
rate is 6% compounded monthly. What s the effective interest rate? If you use the formula
then the quoted rate would be 6% or .06 in decimals. M is 12 because monthly compounded
means compounding means 12 times per year. And your answer will be 6.1678%. In the financial
calculator you do the following. Let’s clear everything we have earlier. Second plus minus
enter. Let’s not worry about changing the decimals, increasing the decimals. Let’s leave
it at 2. So you press second 2. The nominal rate is 6%. You put 6 enter. Press the down
arrow key or up arrow key, either one. Stop when you see compounding per year. You want
to change it to 12. 12 times a year, monthly compounding. 12 enter. Then either up or down
arrow key. Keep pressing. Stop when you see E-F-F effective. Press compute. 6.17% per
year, right? It’s actually rounded to the second decimal place. The more accurate percentage
would be like on my slide 6.1678%. So it’s a little bit below then what the calculator
gave us. The important thing you should realize is that with compounding more than once a
year, effectively you’re earning more money. So let’s say you are saving money in some
bank account. Do you want compounding to be more frequently or less frequent? The answer
should be more frequent because you want to earn as high interest rate as possible. Now
let’s think of a different example. You are taking a loan. You’re borrowing money. Do
you want the interest rate on your loan to be compounded more frequently in one year
or less frequently? The answer this time should be less frequently because you want to pay
a lower interest rate on the money that you borrowed. So it kind o depends which side
you’re on whether you’re a saver or a borrower when the question should compounding be more
frequent or less frequent, right? What is APR annual percentage rates, rate? It’s the
same thing as quoted rate or stated rate or nominal rate, which is how the financial calculator
calls it. This is what the term that banks use. If you go to Bank of America website
or Wells Fargo website they call their interest rates APR. And this is actually the rate banks
are required by law to report to their customers. So you will always see the APR on your checking
account, savings account, credit card and so on. To find the interest rate per each
compounded time interval which si the period rate you take the APR per year and divide
by the number of periods per year. You should never divide the effective rate by the number
of periods per year. It will give you a slightly higher number than the correct interest earned
over each compounded interval. Example a bank charges 1.3% per month on car loans. What
does it mean? What is this bank going to report to customers by law? It’s going to report
the annual interest rate or the APR annual percentage rate of 15.6% which is 12 times
1.3% per month. But if you take a loan to buy a car if you leave a balance unpaid for
one full year are you going to be asked to pay another 15.6% on the amount that you owe?
No it’s going to be more than that. It’s calculated according to the effective annual rate, right?
So how do we get 16.77% per year? We just use the E-A-R formula. By the way banks call
it A-P-Y annual percentage yield. So APR is like the stated quoted nominal rate. APY is
the effective rate. And of course we can verify this result in the financial calculator. Let’s
clear everything first. Second plus minus enter. Let’s increase the decimal places this
time. Second decimal place. Let’s make it 6. Enter. And now let’s convert the APR of
15.6% into the effective annual rate, which should be always a slightly higher number.
Second 2. What’s the nominal rate? 15.6. I press 15.6 enter. Then down arrow key. Down
arrow key again. I stop when I see compounded per year. I change it to 12. 12 enter. Down
arrow key again. One more time. I stop when I see effective and I press compute. 16.765178%
per year. This is the effective rate. If you round it to 2 decimal places you will get
16.77% like on my slide. So this rate the banks are not required to report this higher
rate to their customers even though this is technically the true interest rate that you
owe on your balance after not paying it for one full year. You may or may not seen on
you banks page but banks are only required by law to report the APR, sort of the underestimated
one, the lower one of the two. Let’s look at this example. You deposit \$1,000 in the
bank account with stated annual interest rate of 8% compounded quarterly. How much money
will you have in your account in 5 years? So clearly there’s compounding going on quarterly.
So you probably will need to calculate the effective rate. But this is not what the questions
asking. This actually is a two step problem. In the second step you’re asked to calculate
the future value after 5 years of \$1,000 that you put in your bank account today. So how
should be solve this problem? The first step we calculate by like how may percent per year
does your money grow when we account for the quarterly compounding. And it’s going to a
be a little bit more than 8%. So step one you calculate the effective annual rate. We
can use the formula use 4 for m and 8% for the quarterly rate. That gives us 8.2432%.
And then we use it in the second step to find how much you have your account after 5 years.
So we’re using the effective annual rate to calculate the future value. And the effective
annual rate is going to be our interest rate in the formula. So we have the future value
in 5 years equals \$1,000 multiplied by 1 plus 0.082434. And you close parenthesis and raise
the parenthesis term to the 5. The answer you get is \$1,485.90. This is one way to solve
this problem. And you can do both steps in the financial calculator, right? There’s a
different solution which is a lot easier. You can first realize that we have a total
of 20 quarters over 5 next years. Then every quarter you earn a 2% interest rate on your
money which is 8% per year divided by 4 quarters. And then in the third step you find the future
value after 5 years using 20 quarters for your number of time periods and 2% quarterly
rate for the correct interest rate. And you find the same result. So in the second solution
you skip calculating the effective annual rate completely. You don’t even use it. Example
what is the maximum you should be willing to pay for an investment that pays \$1,000
at the end of each year for 10 years if the interest rate is 5% compounded daily? What
is the main question that we are answered here? The maximum you should be willing to
pay for an investment. So that’s today’s value of some future amounts of money you’ll be
getting back. You’ll be getting back \$1,000 every year for 10 years. So this is an annuity
problem. And we need to find the annuity present value. Okay what are we going to use for the
annuity PMT? \$1,000. What are we going to use for the annuity m? 10 because we have
10 equal payments to us. What are we going to use for the annuity I y interest rate?
5%? No. We need to account for the daily compounded. So it’s actually going to be something a little
bit higher than 5% effectively. How much? Step one calculate effective annual rate using
365 for your compounded frequency. Of course you can use the financial calculator where
the nominal rate is 5. Compounded per year is 365. And you’re computing the effective
rate. We get 5.1267% right here. It’s a little bit higher than the quoted rate of 5%, which
it should always be. And then in the second step you use the effective rate as the interest
rate to find the annuity present value. So your annuity payment is 1,000. Your annuity
interest rate I y is 5.1267 which is the effective rate. And the annuity number of payments is
10 as given. Compute present value. The answer is \$7,674. Let do this in the financial calculator.
Let’s start by clearing everything. Second plus minus enter. Let’s increase the decimal
places. Second decimal place. Let’s make it 6. 6 enter. Step one calculate the effective
rate. You press second 2. The nominal rate which needs to be entered first is 5% quoted
rate which is given. You put 5 enter. Press either up or down arrow key and stop — keep
pressing and stop until you — when you see compounded per year c over y. Change it to
what? Change it to daily compounded. How many times a year is that? 365. So press 3, 6,
5, enter. Well technically there are leap years every four years with 366 but in finance
we just kind of agree to always use 365. Okay so we enter 365. We pressed enter. No it’s
saved. Now you press up or down arrow key. And stop when you see effective on the display.
Press compute. 5.1267 and we actually have a couple more 5, 0, which we don’t have on
the slide. So this is the effective rate. Okay now let’s clear everything again. Let’s
increase the decimals. Second format or decimal place. Let’s make it 6 again. 6 enter. Let’s
start. We need to calculate annuity present value so we’ll be computing present value.
For that we need to enter N-I-Y and P-M-T. What is our m? We have a total of 10 future
payments that will be made to us on the investment we are buying now. So we put 10 n. What is
— how much are we going to be paid every year? \$1,000. Put 1,000 change the sing to
negative and save it as EMT. And last but not least what’s the interest rate? That’s
the effective annual rate from the first step of our calculations. Put 5. Remember we don’t
use decimals. So you need to kind of move the decimal place by 2 to the right. 5.1267.
5.1267. And I think we also had two more numbers when we did it in the calculator. Which were
5, 0. So let’s do that. That’s our interest per year. Compute PB. \$7,674 like on the slide.
And we also have some cents, approximately 43 cents rounded to the, to the second decimal
place. So this a two step problem in which the effective annual rate is the first step
you do. And that’s part of a much bigger problem something like finding annuity present value.
Or a different problem would ask you to fin annuity future value. Or maybe find the present
value of multiple cash flows. Or future value of multiple cash flows. Or something with
perpetuities. In the previous problem what if \$1,000 that we would pay to you every year
on the investment you will be buying now will never end? That’s a perpetuity. Same first
step. Find the effective annual rate with daily compounding. We already know that it’s
5.1267% per year. And in the second step you used this as the discount rate to find the
perpetuity present value. The perpetuity present value formula says take the, take the perpetuity
payment and divide it by the interest rate or discount rate. The perpetuity payment is
\$1,000. The correct discount rate is the effective rate from the first step 5.1267%. The answer
we get is \$19,506. It’s probably rounded a little bit. Example. You manage a local Bank
of America branch. You want to offer a new credit card to your customers that would bring
a 15% return per year. Interest is compounded monthly. What APR will be required to report?
So a card that would bring a 15% return per year from the banks perspective is the true
interest that the banks makes by offering such credit card. So here 15% is actually
the effective annual rate. And the questions asks us to calculate the APR which is the
nominal, the stated quoted annual interest rate. So we need to go back from the effective
to the stated rate using monthly compounding of interest. How do we do that? To go back
from the ER effective rate to APR annual percentage rate we can take the formula we had earlier
for the effective annual rate and rearrange it so we can have APR equals something on
the right hand side. When we rearrange the terms, we get the following; APR equals m
multiplied by in the parenthesis 1 plus ER all together raised to the power 1 over m.
And then we subtract 1 closed parenthesis. In the financial calculator you actually use
the same keys except you enter the effective rate and the compounded per year. And you’re
going to have to compute the number rate. The answer in our problem is 14.06%. So see
how it’s a little bit below the 15% effective which should always be the case. The effective
rate is always higher than the stated rate. So let’s see how the financial calculator
does it. Clear second plus minus enter. Let’s increase the decimals. Second decimal place.
What should we make it? Let’s make it 7 for a change. 7 enter. Now we have 1,2, 3, 4,
5, 6, 7 decimal places. Okay let’s go back from the effective annual rate to the annual
percentage rate. So the number we should get will be smaller than the 15% effective rate
which is given to us. How do we convert interest rates? We press second and 2 as always except
now leave the nominal interest rate information as is. Right away press either the up or the
down arrow key. Keep pressing. Stop when you see effective. This is 15% which is given.
Press 15 then press enter. Then press either up or down arrow key. Stop when you see compounded
per year. We have monthly compounding of interest in our problem. So change this to 12. Press
12 and then press enter. And finally press the up or down arrow key. Stop when you see
nominal. And all you do now is press compute. 14.0579003. So we are keeping 7 decimal places
right? If you round it to 2 decimal places it’s approximately 14.06% like on my slide.
Let’s look at this example that the textbook mentions. Payday loans. What are payday loans?
It’s a very short term loans made to consumers often for less than 2 weeks. So you borrow
money now, you pay it back in 2 weeks. And there are some companies that specialize in
such payday loans such America Cash Advance and National Payday. How do they work? You
walk one of such company’s offices. You say you want to borrow money. You write them a
check today which is post dated in the future. You give it to them and they give you cash.
So you have just borrowed money from them. And on the payday, on your payday when you
know you will have money from your job to pay back the loan the company you borrowed
the money form cashes your check. For example you write the check for 120 dated 15 days
in the future which you know is when you will get your you know money from your employer.
Today these payday loan company gives you \$100 in cash. So you have just borrowed \$100
today and in 15 days you will pay them back 120. So \$20 is basically the interest you’re
paying them. If you go to www.nationalpayday.com the website for one of such companies, you
will see this very happy couple. I wonder why they’re so happy? Because if I were them
I would be definitely crying and you’ll see in a second why. If you click on F-A-Q on
that page it will explain in very small font how they do their math, how much they charge
depending on how much you borrow and for how many days. So for example you can borrow \$100
for 16 days. You will be charged \$25.00 loan fee and so the check you write them that’s
post dated by 16 days should be for the amount \$125. You can also borrow \$200 or \$300. So
it’s all very small amounts of money. That’s what this business is about. You borrow just
a little bit that you need. The fees are pretty high though right? So let’s look at the worst
case scenario. You borrow just a little bit \$100. They charge you a loan fee of \$25 so
after 7 days you will be paying them back \$125. In the last column you will see the
APR corresponding to this case equal to \$1,303 – oh sorry. \$1,303.57%. That’s per year. Let’s
see how they got this number. You borrow 100 they charge you \$25 fee. 7 days later you
pay them back 125. It’s nothing but a 25% interest for 7 days. You know think about
what the interest rate is on let’s say car loans these days. It’s like maybe 4, 5% per
year. This is 25% for 7 days. So it should already look pretty alarming right? Okay what
does it mean in terms of the APR annual percentage rate? What’s the daily rate that you’re being
charged? 25% divided by 7 days that’s per day. And then you multiply by 365 days. And
that’s how we get 1,303.57% per year. But wait a second. This is not even the true interest
rate you’re charged. Image you borrow \$100 and you forget to pay it back for one full
year. You’re not even going to be charged 1303.57% on \$100 that you owe them. You’re
going to be owe — charged even more. So it’s the effective rate that we need to calculate.
What’s the effective rate? The stated rate that goes into the formula or the nominal
rate in the financial calculator is 1,303.57, right? That’s in percent. Compounded per year
is 365 times. You do the math. And you get something unimaginable. The effective annual
interest rate is 36 million 524 thousand 696% per year. It’s probably hard to even to, to
imagine what it really means. But what it means is something like this. If you borrow
\$1 from this company and forget to pay back right? After 7 days. They will come back after
one year. Of course they’re going to wait one year right? They’re going to want to get
as much money from you as they can. So after one year you will owe them \$365,247.96. That’s
from \$1 you borrowed now. So that’s how big of an effect the interest compounded makes
when it’s so often every year. So when its daily may — it you know it’s a big, big difference
right? You know the big difference between the APR the annual percentage rate which is
the quoted rate and the true or the effective rate that you’re charged on your money. Let’s
do this in the financial calculator. Let’s increase the decimal places. Second decimal
place. Let’s make it 6 enter. We want to convert the interest rate from APR to EAR. Press second
2. The nominal rate is 1303.57. 1303.57 enter. Down, down. Compounded per year is 365 times.
3, 6, 5 enter. Down, down. When you see effective you should stop and press compute. 36 million
524 thousand 191373%. It, it’s pretty close to what I have on my slide. So we’re off you
know in the last three digits for the interest rate and in the decimal places. So the reason
why we got the rounding error is because the APR 1303.57% is rounded to the second decimal
place. So if I increase by decimal place before I calculate 1303.57% right? Then I would see
several more numbers after the decimal place. And if I use them of my nominal rate in the
ER interest conversion I would get exactly the number on my slide which is perfectly
accurate which says 36 million 524 thousand 696%.>>On this remaining approximately 15
slides of chapter six I would like to cover different types of loans. There will be some
calculations. But nothing that we have not seen before in chapter six. And I will go
through the calculations, but on the exam they will not be anything requiring math;
so just sort of conceptual problems on these topics. So different loan types. It came up
a few times throughout chapter six that when we deal with loans, when you borrow money
let’s say to buy house or to buy a car or something else then the loan amount should
be viewed as nothing but the present value of all future payments that you will be making
back to your lender. And because all payments are identical this is nothing but an annuity.
So any loan is the present value of annuity of loan payments. In general, we can classify
loans into three different types. First pure discount loans. This means that you borrow
some amount of money today. And after a while, maybe after a year you pay back, so the principle
that you borrowed, you pay back to the lender with some interest on top of it; so it’s interest
rate. And there’s nothing going on between the time you borrow and the time when you
pay the money back; that’s pure discount loans. The second type of loans is interest only
loans. That’s when you do pay something in between, so you pay principle at the end,
so the entire amount that you borrow now you pay that full amount at the very end when
the loan is due. And in between you’re also paying interest; so maybe it’s a three year
loan, they you will pay interest amount in dollars once a year, and the principle at
the very end. The most common type of loans, at least consumer loans are like car loans,
home loans, that people like us will probably deal with at some point in our life are known
as amortized loans or amortizing loans. And this is kind of the most complicated type
out of these three. That’s when each time you make a payment back to the lender you
are paying interest amount and part of the premium; so the fact that you’re paying principle
regularly means that the amount that you owe to the lender goes down with each payment.
So now let’s see how each of these works. Pure discount loans, so again you borrow money
today, you pay back in one payment, which includes the money that you borrowed plus
the interest on top. So a single lump sum payment. And where is this type of loans common?
It’s common for government bonds, which is a way for the US government to borrow money,
also known as treasury bills or T Bills. And the duration of the loan is just a few months;
so under a year, which means the government borrows the money let’s say from you. And
in let’s say 12 months; in one payment it gives that amount back to you with some interest
on top. So let’s look at this example. We have a treasury bill, a T Bill that promises
to repay \$10,000 in 12 months. The market interest rate is 7%. How much will this treasury
bill sell for in the market? In other words, what this question is asking is how much does
the US government borrow with each of these T Bills, right? By borrowing it’s basically
selling it and the money it receives from whoever buys the bond is the loan that was
made to the US government by that buyer of the T Bill. So how much will the bill sell
for in the market? \$10,000 is in 12 months, it’s in the future, so it’s like what will
be paid later. And we need to find how much is borrowed today. And remember what’s repaid
is a single lump sum payment, which includes the amount that was borrowed or in other words
how much the bill sold for today, plus the interest amount on top, so \$10,000 is interest
plus the – how much was borrowed by the government. And the way we calculated it is really, really
simple calculation. So it’s in a way what we were doing in chapter five, going back
to chapter five. You have future value of \$10,000. You want to bring it back by one
year, so we discount it back by one year to find the borrowed amount of money. In the
financial calculator \$10,000 is the future value, IY, interest per year is 7%. And the
number of years is one, because \$10,000 is repaid in one year. We compute the present
value; so the answer is \$9,345.79. That was pure discount loans. Second interest only
loans. These are a little bit more complicated, but not by too much. That’s when you borrow
money today and you pay the principle at the end, just like with the pure discount loans,
but you don’t pay the interest together with the principle in one shot at the end. But
you pay regularly in between. A lot of corporate debt works this way, so corporations take
so-called interest only loans, which require them to pay interest each let’s say year and
they pay back the money that they borrowed with the last interest payment at the end,
maybe in a few years. Let’s see how the math would work here. Let’s say a company borrowed
money for five years. It’s a five year interest only loan. The interest rate is 7% per year.
The principle amount, in other words the amount that the company borrowed is again, \$10,000.
So let’s just use the same numbers as much as we can throughout this entire topic on
loans. Interest is paid annually. The question is what would this stream of cash [Inaudible]
look like? In other words, how much would this company that just took this loan be paying
during the first year, during the second year, third, fourth and fifth years. Right? Here’s
year to date, that’s when we borrowed the money and then we make the payments over the
next five years regularly. And we pay interest every single year, so five times. And at the
end of the fifth year we also pay the principle amount. In other words how much we borrowed
in year zero. So we borrowed what, \$10,000, right? Which is given. What is the interest
that we will be paying every year for the next five years? We use the interest rate
of 7%. Seven percent times \$10,000; that’s how we calculate it, which gives \$700.00 every
year. So year one we pay \$700. Year two we pay \$700. Year three we pay \$700. Year four
we pay \$700. Year five we pay \$700 and we also pay \$10,000 that we borrowed today. So,
the fifth year payment is the largest one because it includes two parts. And it’s a
total of \$10,700. And now let’s look at the most complicated one of all, which is probably
the most common one, amortized loans. That’s when we make payments regularly during the
time when the money is borrowed, and the time when everything must be completed, all the
payments are made. And each of these regular payments includes two things, interest and
principle. Home loans, car loans are examples of where you always see these kind of loans,
this type, the amortized loans. There are several different versions of amortized loans.
Let’s look at A, B and C. The three versions which are discussed in our text book. The
first version, version A, amortized loans where each period, so typically loans require
you to make payments every month. So let’s say every month you pay interest plus some
fixed amount. And this is the key here, fixed amount. Let’s – let’s just jump straight to
the main example I prepared. Let’s make it a \$50,000 loan, right? You borrow \$50,000
for 10 years at 8% annual interest rate. And the loan agreement requires that you pay \$5,000
in principle each year, so this amount is fixed, it doesn’t change, it’s constant. Plus
interest for that year. And again we want to look at what will be going on during the
next 10 years, what will this firms payments back to the lender look like. And typically
what we do is we look at each year separately. For each year we look at the begin balance;
so how much we still owe to the bank. The principle payment that is made during that
year, the interest payment, the total which is the sum of the two and then end of that
year. How much we still owe to the bank. This is what is known as an amortization schedule;
so it’s like a table of payments over the term of the loan. So we can actually click
on this Excel icon. Double click on it. You will see just a second. Something like this.
Let’s make it larger. Okay what’s going on here? We have a table that shows the next
10 years, right? We have beginning balance, principle, interest, total and ending balance.
We are given \$50,000 right, this is the loan amount. And at the end of year 10 we should
owe nothing to the lender, right? So this is zero. So we start with \$50,000 and we should
have zero remaining. What’s going on in between? This is what we were given, right? The principle
is fixed; so we are required to make \$5,000 payments every year for the next 10 years,
right? This is given. But this is not all we will be paying. Otherwise it’s really like
splitting \$50,000 into 10 equal parts over the next 10 years. And no interest, right?
But then realize there’s always interest that you pay to the lender. And the interest rate
is 8%, which is given. So how does this you know, affect our amortization schedule? First,
we can calculate the interest paid during year one. And what we do is actually the same
thing we were doing in the interest only loans. We take 8% interest rate and multiply by \$50,000
that we owe at this point to our bank. So \$50,000 times 8% interest rate gives \$4,000.
Okay? Now what this means is we are paying \$5,000 in principle, another \$4,000 in interest,
which gives a total of \$9,000, right? So this is the total we are paying during year one.
Now what is \$45,000 in the last column? The ending balance. Well this is how much we still
owe to the lender. And to get this number you look at how much you owed as of the beginning
of the year, and how much you paid in principle, not total but principle, which was \$5,000.
So \$50,000 beginning balance, \$5,000 was paid to – so this is by how much our balance has
been reduced over the first year. And now what’s left is \$45,000. And you can actually
notice that because we are reducing the principle by \$5,000 our ending balance goes down by
\$5,000 every year, right? All the way down to zero at the very end. Now what happens
during years two through 10? Actually the same thing. So to find the interest we take
8% and multiply by the balance, right? How much we still owe to the bank. 8% of \$45,000
gives \$3,600. Then to find the total payment that we are making to the lender we combine
the principle and the interest. The principle was given; the interest is based on how much
we owe to the bank. So, a total of \$8,600. And so on, so we kind of – we can keep doing
this, you know for the rest of the years. So what you notice is the beginning balance
goes down by \$5,000 and so does the ending balance, all the way to zero dollars at the
end of 10 years. The principle is fixed, that’s the whole point of this version of amortized
loans. By the way, what does the word amortized mean here? It basically means that we are
slowly lowering the amount we still owe to our lender, right? So it goes down by \$5,000
year after year after year in this particular example. Another thing you can notice is that
interest goes down, and that should make sense because interest is calculated based on how
much we still owe, which is going down. And because the principle is the same, the interest
is going down; the total is also going down just like the interest payment. So this is
the amortization schedule. Another thing I did here on the bottom is to kind of illustrate
how loans are dangerous for you, right? So try to avoid borrowing money as much as you
can. That’s the bottom line. Because even though you borrowed \$50,000, you’re paying
it off in 10 equal components, right? A total of \$50,000 which is the sum of this. But you’re
paying another \$22,000 in interest over the years; so that’s really almost half on top
of your loan, right? So this is the total you are paying over 10 years. \$72,000, right?
So you’re not paying \$50,000; you’re paying \$72,000 out of your pocket. Okay so this is
one version of amortized loans. Now let’s look at version B, where again each time period,
let’s say every month or every year we are paying – making a loan payment back to the
lender, which again includes two components interest and principle. Except what’s fixed
now is the total, the sum of the two. So let’s see what that means. And I already mentioned
that this is the way car loans and home loans and other let’s say consumer loans from Home
Depot, all sorts of consumer loans to purchase electronics and furniture and other things,
that’s how they work. Again, let’s use the same information. You borrow \$50,000. You
need to pay back over 10 years. The interest rate they charge you is 8% per year. We want
to again see what the amortization schedule will look like. But here we are dealing with
fixed annual payments. So let’s click on this Excel icon. It will bring up a slightly different
table, which is actually significantly more complicated. See how all numbers are kind
of you know, there are cents and all sorts of stuff going on. Although the idea is similar.
So the key here is that the total is the same. Every year you’re paying the same total amount,
which is still the sum of principle and interest. The total is the same, but the principle is
very different each year, and interest is different too. So how do we find all this?
\$50,000 is how much we borrow now, right? Every year we are making the same payment.
What – how do we find this number? How do we find that the total payment each year is
\$7,451.47? Well let’s use the help of the financial calculator. What is it that we need
to solving for? What does \$50,000 – remember; any loan is the present value of an annuity;
so let’s start with that. \$50,000 let’s make it negative. And save it as PV, present value.
How many payments are we going to make? Ten, right? It’s a 10 year loan, 10 and then you
press the N button, the number of payments. What is the interest rate that we are given?
It’s 8% per year. That’s our IY, 8 IY. We enter three numbers, which allow us to calculate
the repeating value month, the annuity payment each year or the loan payment, right? So we
are doing compute PMT. \$7,451.47, just like in the column labeled total, every year. So
this is where this number comes from, right? Now what about principle and interest? How
do we split the total into interest and principle? And why is it different every year? So let’s
start you know in the sort of opposite order, let’s look at the interest first. Interest
is always calculated the same way for any type of loan. How do we do it in the earlier
Excel spreadsheet? We said interest in year one is nothing but 8% interest rate, multiplied
by the beginning balance in year one, which is \$4,000. Now we know the total, we know
the interest part of the total. We can then calculate the principle part of the total.
So total minus interest equals principle, right? And now that we know the principle,
we can then calculate the ending balance. So we borrowed \$50,000, we paid a lot of money
during the first year, part of which was the principle amount, \$3,451.47. So this is how
– by how much we have lowered our – like how much we still owe to the lender, right? So
how much do we owe by the end of year one? \$50,000 minus the principle, which is \$46,548.53.
And the ending balance at the end of year one is the same thing as the beginning balance
at the beginning of year two. And then we can redo the whole thing all over again. We
already know the total; interest is 8% of the beginning balance in year two, which gives
\$3,723.88. Total minus interest gives us the principle, which then allows us to calculate
the – again reduced ending balance for the end of year two, and so on. So the total is
the same for each year. The interest is going down because it’s always calculated based
on how much we still owe, which is slowly decreasing over time. Because the total is
the same and the interest is going down, the principle part of the total payment is going
up. But when you do the math, when you add up all the principle payments you should get
exactly the loan amount, \$50,000. And once again you, we noticed that over the 10 years
you’re not going to pay just \$50,000 in cash out of your pocket. You’re going to pay another
almost half of that on top in interest. So you’re going to end up spending close to \$75,000
out of your pocket on a \$50,000 loan. Okay. So that’s how more common amortized loans
work, such as car loans and home loans. Now let’s compare the two versions A and B. A
was the principle part of the total payment was the same every year, case B was the total,
which is interest plus principle was fixed. And the way we did the math was kind of in
– actually exactly you know everything was done in the opposite order. In case A we found
the principle first, which was given. Then interest which is the balance times the interest
rate, and then we add up one and two to get the total. Here in part B, we know – we first
figured out the total based on you know, the financial calculator keys, so that was finding
the payment in an ordinary annuity present value setting. Then we calculated the interest,
then we subtract the interest from the total to find the principle. So it was done in reverse
order. Let’s do another comparison of the two versions of amortized loans. Let’s compare
them by the beginning balance for each year; so year one we start with \$50,000 in both
cases, right? That’s the tallest column on the left on each graph. Then in version A
we were like climbing down stairs, it really looks like a staircase .We are going down
each step has the same height, which was a \$5,000 reduction in the loan balance every
year, right? So it’s like a straight line. Here in part D in the second case we then
looked at its kind of strange staircase, never – it’s not something you see in real life.
So what’s going on here is it’s like bent outwards, right? It’s like a bent out curve
because we saw how in my Excel spreadsheet how when you do the math with the amortization
schedule for each year, you realize that you’re paying more and more and more in principle
every year. So the principle is going up, right? The steps are getting higher and higher
and higher, taller and taller and taller. And in both cases you are paying less and
less in interest with each year, because it’s always based on how much you still owe, which
is going down. We can also compare the two types of amortized loans by how much you are
paying total every year, total annual payment. In the first case with fixed principle payments
the principle is fixed and on top of that we add interest, which is going down over
time. And so the total is also going down over time. In the second version we looked
at with fixed total payments like the name implies, the total is the same it’s flat.
But within the total we can see how the interest is going down, which is actually just like
in case A. And – but the principle is going up over time. And now let’s look at case C,
amortized loans are known as partial amortization loans, also known as balloon loans or bullet
loans. So there’s a payment prior to amortization terms, so prior to the time when the loan
payments are due, like the last one, when it should be made. So sometimes before that
point of time you make like a special payment, a balloon payment. Where do we see balloon
payments? Actually they’re extremely common. Let’s say you buy a house. Typically people
take a loan. Typical loan term is for 30 years, maybe 15, but 30 years is more common. But
nobody lives in a house for 30 years. People move, they find better jobs, right? So let’s
say people move after maybe four years. What happens then? They sell their house, right?
Do they get to keep all the money? No. They give back to their bank whatever they still
owe at that point of time. So they pay back their remaining balance. And only what’s remaining
is theirs to keep. So that last big payment, the entire remaining balance is nothing but
the balloon payment or the bullet payment. There are other cases when like balloon payment
takes place, but let’s not worry about those. So how does it work? So this kind of shows
sort of on the timeline what’s going on? So let’s say you take a loan for 30 years, so
maybe you bought a house. You took a loan for 30 years and so you are supposed to make
fixed total payment every – let’s say every month, right? In real world it’s every month
that you’re making payments on a home loan. But you don’t do that for 30 years. You basically
prepay the loan at some point, much, much sooner. Let’s say you sell your house and
move away to a new house. So there is an amortization term which is used to calculate the required
payment over 30 years or whatever the loan term is. But the amortization term is, but
the actual loan term here is much shorter and the last payment is much larger than all
payments that have been made so far. So that’s the balloon payment. Okay let’s look at the
same problem. You take a loan for \$50,000 for 10 years at 8% interest rate. You’re required
to make fixed annual payments; so essentially this version of amortized loans with a balloon
payment is just like the one we just did with fixed total annual payments, fixed total,
right? In this version, in this problem on this slide, the borrower pays a balloon payment
after four years. So maybe you know it’s a loan to buy a cheap house somewhere for 10
years, although that’s not typical in real life. And after four years we are selling
the house and paying back the remaining balance to the bank. What is the amount of the balloon
payment? So how much are we going to pay to the bank after four years? Actually there
are a couple different ways to do it. So you can either go back to our Excel spreadsheet,
let’s see I’m trying to find – I think go back here in our previous problem because
this is where the answer is. We don’t need to create any new tables. So where is the
answer here? Year four, at the end of year four we are making the last payment to the
bank, right? How much is it? What’s the balloon payment? Here it is. This is our balloon payment,
\$34,447.27. So that’s basically nothing but the ending balance for year four, right? So
essentially in year four your paying this, the balloon payment \$34,000 plus you also
don’t forget that there’s another about \$7,500 on top of it, right? The interest and principle
reduction. So this will be basically the sum of these two, so it’s like \$42,000 roughly
is your last payment during the fourth year. Okay so going back to this slide. I’ve just
shown you how – where to find it in our amortization schedule. So it’s just the ending balance
for that year. But there’s a different way. You can also say that – so let’s say you don’t
have the amortization schedule, you don’t have it built. How else could you find the
balloon payment? The balloon payment basically reflects how much you still owe to your bank.
And it also reflects what’s still remaining. If you’re selling – if you are prepaying the
loan after four years and the loan is originally for 10 then there are six remaining payments.
And we need to essentially add them up, but in the correct way. The way we do it in [Inaudible]
3000 class, in Finance. We find their combined present value, right? So we basically discount
all of them back to end of year four and then add up the six individual present values.
How do we do this? You can use the formula to find the present value of an ordinary annuity
with six payments at 8% interest rate. Each payment being our total for each year. Or,
in the financial calculator you are computing the present value, PV. You’re using the PMT,
payment of almost \$7,500. IY is 8% and the number of payments is six. Six remaining payments
compute PV. So these are kind of you know, all the different ways how you could do it.
Look up amortization schedule if you have it in front of you already built, or use the
formula to find annuity present value or let the financial calculator do it for you. And
here let’s just do it in the financial calculator. Let’s clear everything, start all over again.
Payment is \$7,451.47. Let’s make it negative and the only reason we make it a negative
number is so that when the calculator shows us the answer, it’s shown without any negative
signs. Okay so we want to save it as payment, PMT. The repeating annuity amount. We have
six of those future payments, we press 6N. 8% is the interest rate that we are paying
to the lender, 8 IY and we are computing the present value. \$34,447.25. So I guess here
it’s a little bit rounded, that’s why I’m \$.02 off, but it’s otherwise done correctly.
Okay so we looked at you know, the different loan types. And this is the end of chapter
six.